11institutetext: Centre for mathematical Plasma Astrophysics, Department of
Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001, Leuven, Belgium
22institutetext: Centre for Computational Helio Studies, Ilia State
University, G. Tsereteli street 3, 0162 Tbilisi, Georgia
33institutetext: Evgeni Kharadze Georgian National Astrophysical Observatory,
M. Kostava street 47/57, 0179 Tbilisi, Georgia
44institutetext: Ruhr-Universität Bochum, Institut für Theoretische Physik IV,
44780 Bochum, Germany
55institutetext: Space Research Institute, Austrian Academy of Sciences,
Schmiedlstraße 6, 8042 Graz, Austria
66institutetext: Institute of Physics, University of Maria Curie-Skłodowska,
Pl. M. Curie-Skłodowska 1, PL-20-031 Lublin, Poland

# Wide-band fluctuations of solar active regions probed with SHARP
magnetograms

G. Dumbadze 112233 B.M. Shergelashvili 223344 M.L. Khodachenko and S. Poedts
551166

###### Abstract

Context. The power spectra of the fluctuation noise of the solar active region
(AR) areas and magnetic fluxes sequentially observed in time contain
information about their geometrical features and the related fundamental
physical processes. These spectra are analysed for five different ARs with
various magnetic field structures.

Aims. The goal of this work is to detect the characteristic properties of the
Fourier and wavelet spectra evaluated for the time series of the fluctuating
areas and radial magnetic fluxes of the active regions. Accordingly, this work
gathers information on the properties of noise in the different cases
considered.

Methods. The AR area and radial magnetic flux time series were built using
SHARP magnetogram datasets that cover nearly the entire time of the ARs’
transits over the solar disk. Then we applied Fourier and wavelet analyses to
these time series using apodization and detrendization methods for the cross-
comparison of the results. These methods allow for the detection and removal
of the artefact data edge effects. Finally, we used a linear least-squares
fitting method for the obtained spectra on a logarithmic scale to evaluate the
power-law slopes of the fluctuation spectral power versus frequency (if any).

Results. According to our results, the fluctuation spectra of the areas and
radial magnetic fluxes of the considered ARs differ from each other to a
certain extent, both in terms of the values of the spectral power-law
exponents and their frequency bands.

Conclusions. The characteristic properties of the fluctuation spectra for the
compact, dispersed, and mixed-type ARs exhibit noticeable discrepancies
amongst each other. It is plausible to conclude that this difference might be
related to distinct physical mechanisms responsible for the vibrations of the
AR areas and/or radial magnetic fluxes.

###### Key Words.:

Sun: atmosphere; Sun: corona; Sun: magnetic fields; Methods: data analysis;
Methods: statistical

##  1 Introduction

The solar interior and its atmosphere consist of a complex system of
magnetically shaped, dynamic or static plasma features. The topology of the
magnetic field carries information on the formation of the cyclically variable
large-scale solar magnetic field and its fragmentation into smaller structures
related to sunspots, magnetic loops, filaments, prominences, and so on. On the
other hand, these structures are able to sustain various periodic and
aperiodic, as well as background stochastic processes, of the plasma and
magnetic field motion or modification. Their random nature in many cases
indicates the presence of substantially non-equilibrium random driving
mechanisms (e.g. see Maes et al. 2009).

In our previous studies of solar active regions (ARs), their quasi-periodic
behaviour has been reported in Dumbadze et al. (2017, referred to as Paper I
throughout the text) and in Dumbadze et al. (2021, referred to as Paper II
throughout the text), where different types of AR oscillations with the
characteristic periods of 9-10 h, 6-7 h, and 3.5-4 h have been revealed using
magnetogram datasets of Solar Dynamics Observatory (SDO; Pesnell et al. 2012)
Helioseismic and Magnetic Imager instrument (HMI; Scherrer et al. 2012; Schou
et al. 2012).

These long-period processes seem to be permanently present in the ARs as some
kind of ’undamped’ oscillations on top of visibly manifested background noise.
Because of the relatively low signal-to-noise ratios (S/Ns), special
techniques have been applied to reveal the characteristics of these
oscillations. This behaviour is similar to the recently observationally
detected short-period undamped transverse oscillations in coronal loops
(Nakariakov et al. 2022, 2021, 2016). Although the actual character (e.g.
transverse or longitudinal) of the long-period coherent fluctuations described
in Paper I and Paper II is not yet clear, the similarity of the dynamic
pictures with those taking place for the short-period transverse oscillations
in the coronal loops persists. In addition, the existence of a myriad of
small-scale, stochastically distributed aperiodic motions of EUV intensity
enhancements were recently reported (Shergelashvili et al. 2022). These were
detected and categorised using the statistical categorisation method developed
in Philishvili et al. (2021). It is plausible to believe that the
aforementioned long-period AR oscillations are also strongly dependent on the
background noise. Therefore, the investigation of the noise fluctuation
spectra accompanying the earlier discovered coherent signals is the main
subject of the work presented here.

Auchère et al. (2014) analysed the intensity data cube, $195\;$Å, provided by
the Solar and Heliospheric Observatory (SOHO; Domingo et al. 1995), with the
Extreme ultraviolet Imaging Telescope (EIT; Delaboudinière et al. 1995), and
identified 917 events of solar origin. The intensity pulsations with periods
of 3 to 16 h took place in these events on top of an approximately power-law
distributed background noise spectrum. However, Auchère et al. (2014) could
not identify the physical cause of these pulsations. Froment et al. (2015)
repeated the analysis of Auchère et al. (2014) for the Atmospheric Imaging
Assembly (AIA; Lemen et al. 2012) instrument of SDO and found more than 2000
intensity pulsation events across a similar long-period range. For these
events, they specified three dominant power peaks at around 9.0 h, 5.6 h, and
3.8 h, respectively, which were also addressed in Auchère et al. (2016). In
addition, ultra-long-period oscillations ($\sim 10-30\;$h) were found in the
EUV filaments by Foullon et al. (2004) and Foullon et al. (2009), using the
SOHO/EIT $195\;$Å datasets.

Kolotkov et al. (2017) used the line-of-sight (LOS) SDO/HMI magnetograms to
investigate 13 hours of observational data on small-scale magnetic flux
structures. They performed a spectral analysis, employing the Hilbert-Huang
transform (HHT) technique that decomposed the signal into 11 empirical modes,
where the last one was ”a long-term aperiodic trend of the signal.” According
to Kolotkov et al. (2017), the remaining nine empirical modes were related to
the noisy components and one mode was shown to have a gradually increasing
oscillation amplitude and its oscillation period increased with amplitude from
$\sim 80$ to $230\;$min.” Efremov et al. (2014, 2018) analysed the dynamics of
the magnetic field strength in sunspot SOHO/MDI magnetograms. They suggested
that the detected long-period (periods 10-35 h) oscillations of the magnetic
field in sunspots are related to the oscillations of the sunspots as a whole.

All the above-mentioned significant period peaks are usually embedded in a
strong background noise that is observable in the power spectra of various
frequency bandwidths. Leonardis et al. (2012) used the Hinode Solar Optical
Telescope (SOT) to investigate the power spectral densities of five areas with
quiescent prominences on the northwest solar limb. They obtained power-law
spectra, $W\propto f^{-|m|}$, with an approximate exponent value $|m|\approx
1.2$. Gupta (2014) obtained almost the same values, namely, of $|m|=1.2-1.7$,
where SDO/AIA $171\;$Å and 193 Å images were used and the Fourier power
spectra of the intensity fluctuations at six locations of the south polar
coronal hole were analysed. Ireland et al. (2015) used SDO/AIA $171\;$Å and
$193\;$Å to analysed six hours of image data at four locations of the image
field: a quiet sun region, a region that lies on top of a sunspot, a footpoint
region, and a coronal moss region. The time series of the images revealed
approximately power-law properties with values of $|m|$ in the range of
$1.8-2.3$. Kolotkov et al. (2016) studied the statistical properties of EUV
emission above the centre of a sunspot and above a quiet sun region, using
SDO/AIA $304\;$Å, $171\;$Å and $1600\;$Å data. In both regions, values of
$|m|$ in the range of $0.86-1.33$ were found in the obtained power-law
spectra.

In this paper, we study the power spectra of the background noise using the
SDO/HMI datasets obtained during the transit of ARs over the visible solar
disk. We consider five different ARs and analyse the properties of their
power-law Fourier spectra. The remainder of this paper is organised as
follows. The observations and data processing are described in Sect. 2. The
results and discussions are given in Sect. 3. Finally, the conclusions are
presented in Sect. 4.

##  2 Observations and data processing

Table 1: Selected ARs. The values of longitudes and latitudes indicate the AR location on the solar surface at the beginning and the end moment of the observation period. While the area values are taken at the moment when the centre of mass of the AR is at or very close to the visible central meridian.  AR | Obs. time | Longitude | Latitude |  Area2
---|---|---|---|---
no. | (Hours) |  ($\degree$) |  ($\degree$) |  (km2)
12381 | $240$ |  [$-69.9$ $65.2$] |  [$14.7$ $14.5$] | $1.6\cdot 10^{9}$
12378 | $240$ |  [$-64.5$ $60.2$] |  [$-15.7$ $-14.9$] | $1.5\cdot 10^{8}$
12435 | $225$ |  [$-62.5$ $54.4$] |  [$-15.3$ $-15.4$] | $6.0\cdot 10^{7}$
12437 | $225$ |  [$-57.9$ $59.2$] |  [$-21.6$ $-21.7$] | $6.0\cdot 10^{7}$
12524 | $240$ |  [$-69.2$ $69.6$] |  [$14.9$ $14.3$] | $6.1\cdot 10^{8}$

Time series of the magnetic field radial component $B_{r}$, observed by
SDO/HMI, have been taken from the Space-weather HMI Active Region Patches
(SHARP; Bobra et al. 2014) database. They include rectangular frames that
enclose the corresponding AR and track it along the transit over the visible
solar disk. The tracked AR patches are remapped to a Lambert Cylindrical
Equal-Area (CEA) projection. We considered the same set of five active regions
investigated in our previous study in the Paper-II during their transit over
the visible solar disk (see Table 1 for parameters of the selected ARs). All
the observational features of the considered ARs, which are unimportant for
the present study, can be found in the above mentioned Paper II. We here build
time series of characteristic physical quantities of the ARs using the same
methodology as in Paper II. To investigate the power spectra elaborated from
the data time series for the total cross sectional areas and the corresponding
mean and total magnetic fluxes (while distinguishing between the signed and
unsigned, positive and negative magnetic fields), we perform a Fourier
analysis of each dataset, separately. The particular evaluation methods of the
characteristic physical quantities are outlined in the following subsections.
For clarity and convenience, the acronyms and symbols used are summarised in
Table 2.

Table 2: Summary of the acronyms and symbols used.  acronym | meaning | symbol/def.
---|---|---
CEA | cylindrical equal-area mapping |
| magnetic field threshold | $th$
| frequency | $f$
psd | power spectral densities | $W$
MNMF | mean negative magnetic fluxes | $B_{r-}/N_{th}$
MPMF | mean positive magnetic fluxes | $B_{r+}/N_{th}$
MSMF | mean signed magnetic flux | $B_{r}/N_{th}$
MUMF | mean unsigned magnetic flux | $|B_{r}|/N_{th}$
| number of selected active pixels with $|B_{r}|>\;th$ | $N_{th}$
| size of pixel on solar surface | $S_{CEA}$
| (set by the CEA projection mapping) |
SE | standard error |
TNMF | total negative magnetic fluxes | $B_{r-}$
TPMF | total positive magnetic fluxes | $B_{r+}$
TSMF | total signed magnetic flux | $B_{r}$
TUMF | total unsigned magnetic flux | $|B_{r}|$
| areas of concentration of magnetic field above threshold | $\Sigma_{th}$
| radial magnetic flux in the areas of magnetic field concentration | $\Phi_{th}^{I}$
AVSS | absolute value of the spectrum slope |  $\left|m_{\Sigma_{th}^{J}}\right|$, $\left|m_{\Phi_{th}^{I}}\right|$
| range of AVSS for areas | $\left[\left|m_{\Sigma_{th}^{J}}\right|\right]$
| range of AVSS for radial magnetic fluxes | $\left[\left|m_{\Phi_{th}^{I}}\right|\right]$

###  2.1 The power spectra

Before carrying out a Fourier spectral analysis of the obtained time series,
we performed the standard pre-processing of the data to remove the first,
second, or third-order polynomial trends, where relevant. In addition, we
applied a normalisation procedure in the datasets as
$(X-\overline{X})/\sigma_{X}$, where $X$ is the data count from the
corresponding dataset, whereas $\overline{X}$ and $\sigma_{X}$ are the average
value and standard deviation of $X$, respectively.

To obtain the power spectra, we applied the following procedures. First, we
compute the Fourier spectra to get the power spectral densities (psd), $W,$
out of the normalised datasets. Then, we omit the power of the Nyquist
frequency, as suggested by Vaughan (2005). The power $W$ versus frequency $f$
is plotted in the log-space in the figures throughout the text below. Finally,
we make a linear model fit of the obtained power spectra $W(f)$ in the log-
scale as:

$\lg W_{model}=m\lg f+b,$ (1)

and we determine the coefficients $m$ and $b$ by a least-square fitting
procedure using single-modal linear functions, covering the entire frequency
domain.

To find which part of the dataset satisfies the $H_{0}$ hypothesis, which
implies that the value of the fitted line slope is 0 with respect to a chosen
slope, $m$, we calculate the $t-$ statistics $t_{m}$ for the slope as follows:

$t_{m}=\frac{m}{SE(m)},$ (2)

where

$SE(m)=\frac{\sigma}{\sqrt{\sum_{i=1}^{n}\left(\lg f_{i}-\overline{\lg
f}\right)^{2}}},$ (3) $\overline{\lg f}=\frac{\sum_{i=1}^{n}\lg f_{i}}{n},$
(4) $\sigma^{2}=\frac{\sum_{i=1}^{n}\left(\lg W_{i}-\lg
W_{model,i}\right)^{2}}{n-2},$ (5)

here, $SE(m)$ is the standard error of the slope $m$, $n$ is the total number
of considered frequency samples, $f_{i}$, hence, $n-2$ is the corresponding
number of degrees of freedom as we elaborate two parameter linear regression.
Similarly, for the intercept, we formulate the $H_{0}$ hypothesis that its
value is $b$ and write the corresponding $t-$ statistics $t_{b}$:

$t_{b}=\frac{b}{SE(b)},$ (6)

where

$SE(b)=\sigma\sqrt{\frac{1}{n}+\frac{\overline{\lg
f}^{2}}{\sum_{i=1}^{n}\left(\lg f_{i}-\overline{\lg f}\right)^{2}}},$ (7)

is the standard error of the intercept $b$.

Furthermore, we compare the obtained values of the $t$ statistics with the
corresponding table values of the Student’s $t$-distribution
$t_{(n-2,\alpha/2)}$ for a specific occasion, where $\alpha$ is the size of
area under the right tail of the distribution. We used the values from the
$t$-distribution tables for comparison with the values obtained by $t_{m}$. If

$|t_{m}|<t_{(n-2,\alpha/2)},$ (8)

it means that $H_{0}$ hypothesis is approved and the slope is vanishing
($m=0$) with the significance level $\alpha$. In this way, we find the extreme
high and low frequency bands of the power spectrum, satisfying the $H_{0}$
hypothesis. The hypothesis is rejected if

$|t_{m}|\geq t_{(n-2,\alpha/2)},$ (9)

which is equivalent to the requirement that the statistic’s $p$-value (i.e.
the probability for a random variable to have the Student $t$-distribution
with $n-2$ degrees of freedom is larger than $\left|t_{m}\right|$) satisfies

$p(|t_{m}|)\leq\alpha.$ (10)

This is the criterion for the rejection of the null hypotheses, meaning $m\neq
0$. Therefore, the non-zero $m$ value can be evaluated as the value obtained
by the least-square fitting with the uncertainties written as follows:

$m\pm SE(m)t_{(n-2,\alpha/2)}.$ (11)

A similar $t$-test was carried out for the intercept as well to obtain its
values and error estimations:

$b\pm SE(b)t_{(n-2,\alpha/2)}.$ (12)

Hence, we estimated the slope $m$ and the intercept $b$ for all datasets.

For the AR datasets (areas and fluxes of ARs with the thresholds $\pm 350\;$G,
$\pm 700\;$G and $\pm 900\;$G), we also determined the residuals of the
computed power from the model line Eq. (1) assuming that they follow a second-
order two-dimensional (2D) chi-square ($\chi_{2}^{2}$) distribution as a null
hypothesis. In particular, we tested the goodness of the model fit via a
Kolmogorov-Smirnov (KS) test (Press et al. 2007). When the KS test does not
satisfy the null hypothesis, we adjust the slope, $m,$ and the intercept, $b,$
slightly, until the KS test was fulfilled. So, the obtained model line
indicates for which values of the regression parameters it manifests a well-
defined power-law distribution. However, we do this on cost of sacrificing
slightly the precision of the regression procedure.

Afterwards, we estimated the 95% probability limit $\gamma_{e}$ by assuming
that the noise is 2D $\chi^{2}$ distributed, following Eq. (16) from Vaughan
(2005).

To cross-validate the results, we applied the above described procedures to
four different methods. The first one is method M1, where we analyse the
original data without applying detrendization or apodization procedures (see
Fig. 1, panel M1). The second one is method M2, where the data are pre-
processed with detrendization, namely, a subtraction of the appropriate
polynomial trend from the original data, so that the remaining part of the
data are free of artefacts of the edge effect and satisfy the power-law
dependence proven by the KS test (see Fig. 1 panel M2). The data for method M3
are apodized (see Fig. 1 panel M3) with a Gaussian function with coefficient
$\alpha=3$ (for details see Paper II). In the last method, M4, the data are
apodized and detrended at the same time (see Fig. 1 panel M4). It is important
to note that, for each particular dataset we used the same detrendization
order both in methods M2 and M4. These four methods are generally used in the
entire frequency range of the considered AR fluctuation spectra. However, for
the more detailed study of the middle- and high-frequency part of the spectrum
($\lg f>-1$), to achieve more confidence, we also applied a multi-scale 1D
wavelet analysis of the original data to reveal the background trends (instead
of applying the polynomial fit). We refer to this approach as the WL method.
The comparison of performance of the methods M1, M2, M3, and WL in the reduced
range of frequencies is shown in Fig. 2.

###  2.2 Cross-sectional areas

To evaluate the cross sectional areas, $\Sigma_{th}$, of the ARs, we set the
thresholds, $th,$ for the magnetic field pixel values (similarly to Paper II),
assuming the pixels with a magnetic field strength exceeding the threshold to
be part of the AR. We choose the same values, namely, $th\equiv\\{$
$\pm\leavevmode\nobreak\ 350\;$G, $\pm\leavevmode\nobreak\ 700\;$G and
$\pm\leavevmode\nobreak\ 900\;$G $\\}$, of the thresholds for the magnetic
field in each rectangular snapshot of all five ARs. These thresholds are
necessary for identifying the AR body and distinguishing it from the spurious
small magnetic features around. Hence, the number $N_{th}$ of the selected
’active’ pixels provides an estimate of the total area of the magnetic field
concentration that exceeds the applied threshold $th$. The size of each pixel
on the solar surface is set by the CEA projection mapping as
$S_{CEA}=1.33\times 10^{5}\;$km2 (Bobra et al. 2014). Consequently, the
principal components of the AR image (i.e. the main sunspots or areas of large
magnetic field concentration for the positive and negative polarity) are
defined as:

$\Sigma_{th}=N_{th}S_{CEA}.$ (13)

The uncertainty of the area was estimated by the method given in Saha & Udupa
(2001). Altogether, we considered three basic kinds of the AR cross sectional
areas $\Sigma_{th}$ defined for the positive (i.e. $\Sigma_{th}^{+}$),
negative (i.e. $\Sigma_{th}^{-}$), and total (i.e. $\Sigma_{th}^{tot}$)
magnetic field, specified for all the thresholds introduced above. We defined
the ranges of AR area absolute values of the spectrum slopes (AVSSs)
$\left|m_{\Sigma_{th}^{J}}\right|$ corresponding to various magnetic field
thresholds in Table 3 and Table 3 (see a detailed description of the content
of these tables below, at the start of Sect. 3) defined as:

$\left[\left|m_{\Sigma_{th}^{J}}\right|\right]=\left[\min\left\\{\left|m_{\Sigma_{th}^{J}}\right|\right\\},\max\left\\{\left|m_{\Sigma_{th}^{J}}\right|\right\\}\right],$
(14)

where, $\min$ and $\max$ denote the minimum and maximum values for a given
subset of $\Sigma_{th}^{I}$-s composed by certain combinations of the magnetic
field polarity ($J\subset\\{+,-,tot\\}$) and applied threshold
($th\subset\\{350,700,900\\}$) values. Values within squared brackets indicate
the range of AVSSs, whereas the corresponding subset of considered cases are
within curly brackets.

###  2.3 Mean and total radial magnetic fluxes

We estimated the total (and mean) radial magnetic flux of the considered ARs
in Mx. The radial magnetic flux data give the total signed (TSMF) and unsigned
magnetic fluxes (TUMF) as well as the mean signed (MSMF) and unsigned (MUMF)
magnetic fluxes. Also, we evaluated the total (TPMF) and mean (MPMF) positive
magnetic fluxes as well as the total (TNMF) and mean (MNMF) negative magnetic
fluxes. The combined expression of the various versions of the radial magnetic
flux reads as follows:

$\Phi_{th}^{I}=\sum_{\hbox{\tiny active pixels}}B_{I}(x,y)S(x,y)=$
$=S_{CEA}\sum\left\\{B_{I}(x,y)\;|\;B_{I}(x,y)>th\right\\},$ (15)

where the index $th$ again indicates the applied magnetic field threshold and
the index $I$ indicates the type of the considered magnetic field, $B_{I}$,
for which the flux is defined as follows:

$\displaystyle B_{I}\subset$ $\displaystyle\\{B_{r}\leavevmode\nobreak\
\mbox{(i.e. TSMF)},\quad\;\;\qquad|B_{r}|\leavevmode\nobreak\ \mbox{(i.e.
TUMF)},$ $\displaystyle B_{r}/N_{th}\leavevmode\nobreak\ \mbox{(i.e.
MSMF)},\qquad|B_{r}|/N_{th}\leavevmode\nobreak\ \mbox{(i.e. MUMF)},$
$\displaystyle B_{r+}\leavevmode\nobreak\ \mbox{(i.e. TPMF)},\quad\;\;\qquad
B_{r+}/N_{th}\leavevmode\nobreak\ \mbox{(i.e. MPMF)},$ $\displaystyle
B_{r-}\leavevmode\nobreak\ \mbox{(i.e. TNMF)},\quad\;\qquad
B_{r-}/N_{th}\leavevmode\nobreak\ \mbox{(i.e. MNMF)}\\},$

with the signs $+$ and $-$ in the index denoting the positive and negative
polarity of the magnetic field, respectively. The values $x$ and $y$ in Eq.
(15) are the rectangular coordinates of the active pixels. We employed the
definition via Eq. (15) for all the considered thresholds $th$. Furthermore,
we defined the structuring of the AVSSs of the radial magnetic fluxes,
$\left|m_{\Phi_{th}^{I}}\right|$, and their ranges,
$\left[\left|m_{\Phi_{th}^{I}}\right|\right]$, in a similar manner as done for
the AR areas by Eq. (14):

$\left[\left|m_{\Phi_{th}^{I}}\right|\right]=\left[\min\left\\{\left|m_{\Phi_{th}^{I}}\right|\right\\},\max\left\\{\left|m_{\Phi_{th}^{I}}\right|\right\\}\right],$
(16)

where, $\min$ and $\max$ denote the minimum and maximum values for a given
subset of $\Phi_{th}^{I}$-s composed of certain combinations of the magnetic
field type
($I\subset\\{\mbox{TSMF},\mbox{TUMF},\mbox{MSMF},\mbox{MUMF},\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$)
and applied threshold ($th\subset\\{350,700,900\\}$) values.

Figure 1: Comparative performance of the methods M1, M2, M3, and M4, providing
the fluctuation spectra of area $\Sigma_{350}$ for AR 12524 in the entire
frequency range. The particular case of threshold of $th=350\;$G is shown as
an example. Here, the blue solid line indicates the model line with slope $m$
that is computed by least square fitting (Eq. (1)) to the obtained power
spectrum and the 95% confidence border is represented by the blue dashed line.
Figure 2: Spectra corresponding to case shown in Fig. 1, obtained via approach
2.

##  3 Results

The comparison of the method M1 with the other used methods reveals that the
spectra obtained with this method are drastically influenced by the dataset
edge effects, which presumably create very low-frequency artefacts at the left
edges of spectra constructed via approach 1. In other methods, this problem is
mainly solved. Further on, we denote as approach 1 the analysis based on the
methods M2, M3, and M4, which include the dedicated apodization and/or
detrendization procedures. This approach was applied across the entire
frequency range (the ranges of spectral slopes obtained with approach 1 are
given in Table 3, with example spectra shown in Fig. 1). Approach 2, applied
in the middle-to-high frequency range and, in addition to the M1, M2m, and M3
methods, also includes the method WL (with the ranges of spectral slopes
obtained with approach 2 are given in Table 3 and example spectra shown in
Fig. 2). The latter method employs the multi-scale wavelet analysis for
revealing and exclusion of the background trends. Using either of these
approaches enables identification and extraction of the mentioned low-
frequency (presumably of artificial power) vibrations and also constructing
more precise Fourier or global wavelet spectra of the remaining noise. This is
necessary to obtain a realistic picture describing the nature of the
fluctuation noise. In the consideration of outcomes with respect to approaches
1 and 2 below, for the sake of comparison, we will also refer to the results
of M1 known to be artefact contaminated. However, the scientific conclusions
will be drawn on the basis of approaches 1 and 2 (i.e. M2, M3, M4, or WL) in
the entire or middle-to-high frequency ranges, respectively.

In the subsections of Appendix A, we describe in detail the performed analysis
of the fluctuation noise spectra for each of the considered ARs. The summary
of the revealed absolute values of the AVSSs for the fluctuations of all the
above introduced parameters of ARs (i.e. area, TSMF, TUMF, MSMF, MUMF, TPMF,
MPMF, TNMF, and MNMF). We elaborate on the different ranges of AVSSs for the
constructed subsets of AR areas
$\left[\left|m_{\Sigma_{th}^{J}}\right|\right]$ (defined by Eq. (14)) and
magnetic field fluxes $\left[\left|m_{\Phi_{th}^{I}}\right|\right]$ (defined
by Eq. (16)). In Table 3 (approach 1) and in Table 3 (approach 2), we also
demonstrate some of the relevant subset ranges of the AVSSs, using the same
definitions. In the lower-right fields of these tables, the values are absent
as corresponding ARs are unipolar and, therefore, corresponding ranges of the
values for these ARs, shown in rows 7, 8, and 9 are the same as in rows 4, 5,
and 6, accordingly.

Table 3: AVSS ranges for some important subgroups of cases, revealed using
approach 1. The colours of the AVSSs correspond to the applied magnetic field
thresholds: $th=350\;$G (black); $th=700\;$G (blue); $th=900\;$G (red).

ccccccc N AR 12381 AR 12378 AR 12524 AR 12435 AR 12437
$1^{{\textsuperscript{*}}}$ $\left[\left|m_{\Sigma_{350}^{J}}\right|\right]$
[$1.1\pm 0.11$, $1.4\pm 0.11$] [$1.1\pm 0.12$, $1.3\pm 0.12$] [$1.1\pm 0.11$,
$1.2\pm 0.12$] [$0.8\pm 0.13$, $1.1\pm 0.13$] [$0.9\pm 0.12$, $1.0\pm 0.12$]

$2^{{\textsuperscript{*}}}$ $\left[\left|m_{\Sigma_{700}^{J}}\right|\right]$
[$1.3\pm 0.11$, $1.6\pm 0.11$] [$1.1\pm 0.12$, $1.3\pm 0.13$] [$1.2\pm 0.12$,
$1.5\pm 0.12$] [$1.0\pm 0.12$, $1.1\pm 0.12$] [$1.2\pm 0.11$, $1.3\pm 0.12$]

$3^{{\textsuperscript{*}}}$ $\left[\left|m_{\Sigma_{900}^{J}}\right|\right]$
[$1.5\pm 0.11$, $1.8\pm 0.12$] [$1.3\pm 0.11$, $1.5\pm 0.12$] [$1.3\pm 0.11$,
$1.6\pm 0.11$] [$1.2\pm 0.12$, $1.2\pm 0.12$] [$1.1\pm 0.12$, $1.5\pm 0.11$]
$4^{{\textsuperscript{**}}}$ $\left[\left|m_{\Phi_{350}^{I}}\right|\right]$
[$1.2\pm 0.11$, $1.6\pm 0.12$] [$1.2\pm 0.12$, $1.3\pm 0.12$] [$1.1\pm 0.11$,
$1.2\pm 0.11$] [$0.9\pm 0.13$, $1.2\pm 0.13$] [$0.9\pm 0.13$, $1.1\pm 0.12$]

$5^{{\textsuperscript{**}}}$ $\left[\left|m_{\Phi_{700}^{I}}\right|\right]$
[$1.4\pm 0.11$, $2.0\pm 0.12$] [$1.1\pm 0.11$, $1.5\pm 0.12$] [$1.2\pm 0.12$,
$1.6\pm 0.11$] [$1.0\pm 0.12$, $1.2\pm 0.12$] [$1.2\pm 0.13$, $1.4\pm 0.12$]

$6^{{\textsuperscript{**}}}$ $\left[\left|m_{\Phi_{900}^{I}}\right|\right]$
[$1.6\pm 0.11$, $2.0\pm 0.12$] [$1.3\pm 0.11$, $1.5\pm 0.12$] [$1.4\pm 0.12$,
$1.8\pm 0.12$] [$1.2\pm 0.12$, $1.3\pm 0.11$] [$1.1\pm 0.12$, $1.5\pm 0.11$]
$7^{{\textsuperscript{***}}}$ $\left[\left|m_{\Phi_{350}^{I}}\right|\right]$
[$1.1\pm 0.12$, $1.4\pm 0.13$] [$1.1\pm 0.12$, $1.3\pm 0.13$] [$1.1\pm 0.11$,
$1.3\pm 0.12$] – –

$8^{{\textsuperscript{***}}}$ $\left[\left|m_{\Phi_{700}^{I}}\right|\right]$
[$1.3\pm 0.11$, $1.8\pm 0.11$] [$1.1\pm 0.12$, $1.3\pm 0.12$] [$1.2\pm 0.12$,
$1.5\pm 0.12$] – –

$9^{{\textsuperscript{***}}}$ $\left[\left|m_{\Phi_{900}^{I}}\right|\right]$
[$1.5\pm 0.11$, $1.9\pm 0.11$] [$1.3\pm 0.12$, $1.6\pm 0.11$] [$1.3\pm 0.12$,
$1.8\pm 0.12$] – –

  * *

$J\subseteq\\{+,-,tot\\}$.

  * **

$I\subseteq\\{\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$.

  * ***

$I\subseteq\\{\mbox{TSMF},\mbox{TUMF},\mbox{MSMF},\mbox{MUMF}\\}$.

Table 4: Same as in Table 3, obtained by approach 2.

ccccccc N AR 12381 AR 12378 AR 12524 AR 12435 AR 12437
$1^{{\textsuperscript{*}}}$ $\left[\left|m_{\Sigma_{350}^{J}}\right|\right]$
[$0.9\pm 0.16$, $1.3\pm 0.15$] [$0.9\pm 0.16$, $1.2\pm 0.17$] [$1.0\pm 0.15$,
$1.3\pm 0.16$] [$0.7\pm 0.17$, $1.1\pm 0.17$] [$0.8\pm 0.17$, $0.9\pm 0.17$]

$2^{{\textsuperscript{*}}}$ $\left[\left|m_{\Sigma_{700}^{J}}\right|\right]$
[$1.1\pm 0.15$, $1.4\pm 0.15$] [$0.9\pm 0.15$, $1.3\pm 0.16$] [$1.1\pm 0.15$,
$1.4\pm 0.15$] [$1.0\pm 0.16$, $1.0\pm 0.16$] [$1.1\pm 0.15$, $1.4\pm 0.17$]

$3^{{\textsuperscript{*}}}$ $\left[\left|m_{\Sigma_{900}^{J}}\right|\right]$
[$1.4\pm 0.15$, $1.7\pm 0.17$] [$1.1\pm 0.16$, $1.5\pm 0.14$] [$1.2\pm 0.16$,
$1.5\pm 0.16$] [$1.0\pm 0.16$, $1.2\pm 0.17$] [$0.9\pm 0.16$, $1.4\pm 0.18$]
$4^{{\textsuperscript{**}}}$ $\left[\left|m_{\Phi_{350}^{I}}\right|\right]$
[$1.0\pm 0.14$, $1.6\pm 0.16$] [$1.0\pm 0.16$, $1.2\pm 0.16$] [$1.0\pm 0.15$,
$1.3\pm 0.15$] [$0.7\pm 0.16$, $1.2\pm 0.16$] [$0.8\pm 0.16$, $1.0\pm 0.17$]

$5^{{\textsuperscript{**}}}$ $\left[\left|m_{\Phi_{700}^{I}}\right|\right]$
[$1.2\pm 0.15$, $1.9\pm 0.19$] [$1.0\pm 0.15$, $1.4\pm 0.15$] [$1.1\pm 0.15$,
$1.5\pm 0.14$] [$1.0\pm 0.17$, $1.1\pm 0.16$] [$1.1\pm 0.15$, $1.4\pm 0.16$]

$6^{{\textsuperscript{**}}}$ $\left[\left|m_{\Phi_{900}^{I}}\right|\right]$
[$1.4\pm 0.15$, $2.0\pm 0.16$] [$1.1\pm 0.16$, $1.5\pm 0.14$] [$1.2\pm 0.17$,
$1.6\pm 0.16$] [$1.1\pm 0.16$, $1.2\pm 0.18$] [$0.9\pm 0.16$, $1.4\pm 0.17$]
$7^{{\textsuperscript{***}}}$ $\left[\left|m_{\Phi_{350}^{I}}\right|\right]$
[$0.8\pm 0.16$, $1.4\pm 0.16$] [$0.9\pm 0.17$, $1.2\pm 0.16$] [$0.9\pm 0.16$,
$1.3\pm 0.16$] – –

$8^{{\textsuperscript{***}}}$ $\left[\left|m_{\Phi_{700}^{I}}\right|\right]$
[$1.2\pm 0.15$, $1.8\pm 0.15$] [$1.0\pm 0.15$, $1.3\pm 0.16$] [$1.2\pm 0.16$,
$1.4\pm 0.17$] – –

$9^{{\textsuperscript{***}}}$ $\left[\left|m_{\Phi_{900}^{I}}\right|\right]$
[$1.4\pm 0.15$, $1.9\pm 0.15$] [$1.2\pm 0.16$, $1.6\pm 0.15$] [$1.2\pm 0.15$,
$1.7\pm 0.16$] – –

  * *

$J\subseteq\\{+,-,tot\\}$.

  * **

$I\subseteq\\{\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$.

  * ***

$I\subseteq\\{\mbox{TSMF},\mbox{TUMF},\mbox{MSMF},\mbox{MUMF}\\}$.

The results of performed analysis of the fluctuation noise spectra presented
in this work can be summarised as follows:

  * •

For the AR areas fluctuations:
(A) The specific feature found for the area fluctuation spectra of the compact
AR 12381 is that the AVSSs of the negative magnetic field area are always
larger than those in the case of the positive magnetic field. Such an
interrelation between these features is not seen in the spectra of the
dispersed, mixed, and unipolar ARs studied (see subsection A.1 point (i)).
(B) In the compact AR 12381, the AVSSs ranges are relatively wider as compared
to other considered cases, and they strongly depend on the applied magnetic
field threshold (see subsection A.1 points (i) and (ii)). However, this
dependence on the magnetic field threshold is weaker for all other ARs.

The AVSSs for the dispersed type AR 12378 coincide with those for the compact
AR 12381 in the case of the relatively small magnetic fields threshold
($350\;$G); while for the thresholds ($700\;$G and $900\;$G), they become
significantly smaller (see subsection A.2 points (i) and (ii)).

The mixed AR 12524 comprises the properties of both the compact and dispersed
ARs with the maximal AVSSs so that their ranges for the applied large magnetic
field thresholds somewhat exceed the corresponding values measured for the
dispersed AR 12378. At the same time, they still remain significantly smaller
than those for the compact AR 12381 (see subsection A.3 points i and ii).

The character of area fluctuations in the case of two unipolar ARs, namely, AR
12435 and AR 12437, is similar to that of the dispersed AR 12378. Moreover,
the narrowing of the AVSSs’ ranges, depending on the applied magnetic field
threshold, is more pronounced for the large thresholds (see Section A.4 points
i and ii).
(C) The ranges of AVSSs for the AR area fluctuation spectra, obtained with the
$350\;$G threshold, are close to those typical for the $1/f-$ type spectrum.
This is the case for all ARs under study (see subsections A.1, A.2, A.3, and
A.4 points (ii)).

For the higher magnetic field thresholds applied, the AR area fluctuation
noise spectra for the dispersed bipolar (AR 12378) and two dispersed unipolar
(AR 12435 and AR 12437) ARs remain predominantly close to the $1/f-$ type one.
At the same time, for the compact AR 12381 and the mixed-type AR 12524, the AR
area fluctuation noise spectra manifest the presence of a pink noise for the
$700\;$G threshold, whereas for the $900\;$G threshold, they become inclined
towards the random walk red noise type.

Altogether, the systematic interrelation between the AVSSs of the AR area
fluctuations for the positive, negative, and total magnetic fields does not
depend on the considered total area of the corresponding AR parts, nor on
which parts are leading or trailing. The observed interrelations between the
AVSSs in the case of compact AR 12381 might be the result of its specific
internal magnetic field structure, but this assumption needs further studies.

  * •

For the AR magnetic flux fluctuations:
(A) Similarly to the ARs’ area fluctuations, the AVSSs for the negative
magnetic flux in the compact AR always exceed those for the positive magnetic
flux (see Section A.1, point iii). At the same time, in all the other ARs such
a systematic relation between the corresponding AVSSs is not observed (see
Sections A.2, A.3, and A.4, point iii).
(B) Similarly to the case of the area fluctuation spectra, the ranges of AVSS
for the magnetic fluxes of the compact AR 12381 are relatively wide and
strongly dependent on the applied magnetic field threshold. However, in this
case, the AVSSs for the magnetic fluxes of different polarities are larger
than those for the corresponding area fluctuation spectra (see subsection A.1
point (v)).

For the fully or partially dispersed ARs, the AVSS ranges are rather narrow
and the observed dependence on the magnetic field threshold is weakly
pronounced as compared to the case of the compact AR 12381 (see Sections A.2,
A.3, and A.4 points (v)). In the latter cases, the estimated AVSSs for the
magnetic fluxes of the different polarities are evaluated within the ranges
with widths comparable to those in the cases of the area fluctuation spectra.
However, the difference between them consists of the different maximum and
minimum values of the AVSSs. Although the mixed AR 12524 spectra partially
comprise the features of the compact and dispersed ARs.
(C) For the compact AR 12381 (see the snapshots of this AR in left panel of
Fig. 3), the pink spectrum is observed for the threshold of $350\;$G, with the
AVSSs values still close to the $1/f-$ type spectrum. Nevertheless, they are
not of a strictly $1/f-$ type, as in the case of the area fluctuations. With
regard to the $700\;$G and $900\;$G thresholds, the corresponding AVSSs are
also typical for the pink spectrum, with more inclination towards the red one.
This might indicate the presence of a random walk component in the magnetic
flux fluctuations (see Section A.1, point v).

In the dispersed AR 12378 (see the snapshots of this AR in right panel of Fig.
3) and the unipolar AR 12435 and AR 12437, we have strictly defined $1/f-$type
spectra for all magnetic field thresholds (see Sections A.2 and A.4, points
v). The situation is similar to the case of the mixed AR 12524. However, the
typical values of AVSSs are slightly larger, which is probably due to the
presence of a compact magnetic kernel in this AR (see Section A.3, point v).
(D) The general result for the compact AR 12381 is that the AVSS ranges of the
signed and unsigned magnetic flux fluctuations are wider than those of the
spectra of different polarities, but still strongly dependent on the applied
magnetic field threshold, similarly to other cases of spectra constructed for
this AR. Again, the estimated AVSSs for the signed and unsigned magnetic
fluxes are (in most cases) larger than those for the corresponding area
fluctuation spectra (see Section A.1, point vi). In addition, in the compact
AR, the AVSSs for the unsigned magnetic flux are always larger than those of
the signed (uncompensated) magnetic flux (see Section A.1, point iv). In all
other ARs, we do not observe such a feature (see subsections A.2, A.3, and A.4
points (iv)).

In the dispersed AR 12378 and the mixed AR 12524, it is noticeable that the
AVSSs obtained for the 700 G and 900 G thresholds for the total unsigned
magnetic flux (triangles) appear to be larger than the AVSSs in most of the
other spectra. This might indicate that a random walk process of the magnetic
flux emergence or cancellation is more strongly present in the ARs of these
types.

Figure 3: Example illustration of Helioseismic and Magnetic Imager (HMI)
magnetogram snapshots of the compact (AR 12381) and dispersed (AR 12378)
active regions.

##  4 Conclusions

The case study performed in this work for the ARs’ areas and magnetic flux
fluctuation spectra leads us to the conclusion that they are different for
compact and dispersed type ARs, which can be an indication of the dependence
of these spectra on the distinct spatially random internal structure and
topology of the magnetic field. However, the fact that the AVSSs for the
magnetic fluxes are somewhat larger than the corresponding values for the ARs’
area fluctuations may manifest the presence of the processes (presumably of a
statistically non-equilibrium nature) of the magnetic flux emergence or
cancellation. The evident differences between the area and magnetic flux
spectra can indicate that the background noise of these physical quantities
may be driven by the spatiotemporal processes of different physical natures.
Further analytical modelling and data analysis procedures are needed to
improve our understanding of the relative importance between spatially random
magnetic field structure and temporarily stochastic magnetic flux variability
in the formation of the observed fluctuation spectra.

###### Acknowledgements.

We thank the referee for constructive remarks that led to a significant
improvement of the content. The work of G.D. was partially supported under the
DAAD-SRNSFG Scholarships for Young Scientists, 2023 provided by the Deutscher
Akademischer Austauschdienst (DAAD) and Shota Rustaveli National Science
Foundation of Georgia (SRNSFG) - project Ref. ID. 91881041. We also
acknowledge partial financial support for B.S. from the Ruhr-Universität
Bochum (RUB) with in the VIP programme and from the Deutscher Akademischer
Austauschdienst (DAAD) within EU fellowships for Georgian researchers, 2023
(57655523) - project Ref. ID. 91862684. SP acknowledges support from the
projects C14/19/089 (C1 project Internal Funds KU Leuven), G.0B58.23N and
G.0025.23N (WEAVE) (FWO-Vlaanderen), 4000134474 (SIDC Data Exploitation, ESA
Prodex-12), and Belspo project B2/191/P1/SWiM.

## References

  * Auchère et al. (2014) Auchère, F., Bocchialini, K., Solomon, J., & Tison, E. 2014, A&A, 563, A8
  * Auchère et al. (2016) Auchère, F., Froment, C., Bocchialini, K., Buchlin, E., & Solomon, J. 2016, ApJ, 825, 110
  * Bobra et al. (2014) Bobra, M. G., Sun, X., Hoeksema, J. T., et al. 2014, Sol. Phys., 289, 3549
  * Delaboudinière et al. (1995) Delaboudinière, J. P., Artzner, G. E., Brunaud, J., et al. 1995, Sol. Phys., 162, 291
  * Domingo et al. (1995) Domingo, V., Fleck, B., & Poland, A. I. 1995, Sol. Phys., 162, 1
  * Dumbadze et al. (2017) Dumbadze, G., Shergelashvili, B. M., Kukhianidze, V., et al. 2017, A&A, 597, A93
  * Dumbadze et al. (2021) Dumbadze, G., Shergelashvili, B. M., Poedts, S., et al. 2021, A&A, 653, A39
  * Efremov et al. (2014) Efremov, V. I., Parfinenko, L. D., Solov’ev, A. A., & Kirichek, E. A. 2014, Sol. Phys., 289, 1983
  * Efremov et al. (2018) Efremov, V. I., Solov’ev, A. A., Parfinenko, L. D., et al. 2018, Ap&SS, 363, 61
  * Foullon et al. (2004) Foullon, C., Verwichte, E., & Nakariakov, V. M. 2004, A&A, 427, L5
  * Foullon et al. (2009) Foullon, C., Verwichte, E., & Nakariakov, V. M. 2009, ApJ, 700, 1658
  * Froment et al. (2015) Froment, C., Auchère, F., Bocchialini, K., et al. 2015, ApJ, 807, 158
  * Gupta (2014) Gupta, G. R. 2014, A&A, 568, A96
  * Ireland et al. (2015) Ireland, J., McAteer, R. T. J., & Inglis, A. R. 2015, ApJ, 798, 1
  * Kolotkov et al. (2016) Kolotkov, D. Y., Anfinogentov, S. A., & Nakariakov, V. M. 2016, A&A, 592, A153
  * Kolotkov et al. (2017) Kolotkov, D. Y., Smirnova, V. V., Strekalova, P. V., Riehokainen, A., & Nakariakov, V. M. 2017, A&A, 598, L2
  * Lemen et al. (2012) Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17
  * Leonardis et al. (2012) Leonardis, E., Chapman, S. C., & Foullon, C. 2012, ApJ, 745, 185
  * Maes et al. (2009) Maes, C., Netočný, K., & Shergelashvili, B. M. 2009, Phys. Rev. E, 80, 011121
  * Nakariakov et al. (2021) Nakariakov, V. M., Anfinogentov, S. A., Antolin, P., et al. 2021, Space Sci. Rev., 217, 73
  * Nakariakov et al. (2016) Nakariakov, V. M., Anfinogentov, S. A., Nisticò, G., & Lee, D. H. 2016, A&A, 591, L5
  * Nakariakov et al. (2022) Nakariakov, V. M., Kolotkov, D. Y., & Zhong, S. 2022, MNRAS, 516, 5227
  * Pesnell et al. (2012) Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, Sol. Phys., 275, 3
  * Philishvili et al. (2021) Philishvili, E., Shergelashvili, B. M., Buitendag, S., et al. 2021, A&A, 645, A52
  * Press et al. (2007) Press, W., Teukolsky, S., Vetterling, W., & Flannery, B. 2007, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University Press)
  * Saha & Udupa (2001) Saha, P. & Udupa, J. 2001, IEEE Transactions on Pattern Analysis and Machine Intelligence, 23, 689
  * Scherrer et al. (2012) Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207
  * Schou et al. (2012) Schou, J., Scherrer, P. H., Bush, R. I., et al. 2012, Sol. Phys., 275, 229
  * Shergelashvili et al. (2022) Shergelashvili, B. M., Philishvili, E., Buitendag, S., Poedts, S., & Khodachenko, M. 2022, A&A, 662, A30
  * Vaughan (2005) Vaughan, S. 2005, A&A, 431, 391

##  Appendix A Detailed description of the obtained spectra

###  A.1 AR 12381

AR 12381 has a compactly organised magnetic field that appears in two main
large sunspots, where the leading and trailing ones have the negative and
positive polarities, respectively. The following features of the AR 12381
fluctuation noise spectra can be distinguished.

(i) The AVSSs of the AR area fluctuations for the negative magnetic field
$\left|m_{\Sigma_{th}^{-}}\right|$ (Fig. 4 (a), (b), pluses) are
systematically larger than those for the positive magnetic field
$\left|m_{\Sigma_{th}^{+}}\right|$ (Fig. 4 (a),(b), crosses). Here and further
on, if the particular value of the applied magnetic field threshold $th$,
magnetic field polarity and type or applied approach of analysis is not
specified, then we assume that the corresponding statement holds true for all
the thresholds, magnetic field cases or both approaches, accordingly.

(ii) The spectral analysis (Fig. 4, a) shows that the AVSSs for the full set
of AR areas, $\left|m_{\Sigma_{th}^{J}}\right|$,
$th\subseteq\\{350,700,900\\}$, $J\subseteq\\{+,-,tot\\}$, fluctuation noise
range within [$1.1\pm 0.11$, $1.8\pm 0.12$] by approach 1, while approach 2
(Fig. 4, b) yields the AVSSs within [$0.9\pm 0.16$, $1.7\pm 0.17$]. The AVSSs
for the area fluctuations systematically increase with the applied magnetic
field threshold $th$ in this AR (see Table 3 and 3 rows 1,2,3). Accordingly,
for $th=350$ G, the spectrum is close to the $1/f-$ type one, with ranging
within $\left[\left|m_{\Sigma_{350}^{J}}\right|\right]$ = [$0.9\pm 0.16$,
$1.4\pm 0.11$], $J\subseteq\\{+,-,tot\\}$ (black marks). For $th=700$ G
threshold, the well manifested pink noise spectrum with ranging within
$\left[\left|m_{\Sigma_{700}^{J}}\right|\right]$ = [$1.1\pm 0.15$, $1.6\pm
0.11$], $J\subseteq\\{+,-,tot\\}$ (blue marks) is seen. Application of the
threshold $th=900$ G yields the spectra with power-law inclined towards the
red noise type with ranging within
$\left[\left|m_{\Sigma_{900}^{J}}\right|\right]$ = [$1.4\pm 0.15$, $1.8\pm
0.12$], $J\subseteq\\{+,-,tot\\}$ (red marks).

(iii) As shown in Fig. 4 (c) and (d), the AVSSs in this AR, for both, the
total $\left|m_{\Phi_{th}^{TNMF}}\right|$ (triangles) and mean
$\left|m_{\Phi_{th}^{MNMF}}\right|$ (circles) negative magnetic fluxes of the
leading spot are in the respective order larger than those for the total
$\left|m_{\Phi_{th}^{TPMF}}\right|$ (diamonds) and mean
$\left|m_{\Phi_{th}^{MPMF}}\right|$ (squares) positive magnetic fluxes in the
trailing spot.

(iv) Similarly to the case of different polarities, the AVSSs of both the
unsigned total $\left|m_{\Phi_{th}^{TUMF}}\right|$ (triangles) and mean
$\left|m_{\Phi_{th}^{MUMF}}\right|$ (circles) magnetic flux spectra are also
almost always (the exception is the case of $th=900$ G in method M3) higher
when respectively compared to those of the signed total
$\left|m_{\Phi_{th}^{TSMF}}\right|$ (diamonds) and mean
$\left|m_{\Phi_{th}^{MSMF}}\right|$ (squares) magnetic flux (Fig. 4 (e) and
(f)).

(v) The spectral analysis by approach 1 (Fig. 4, c)shows that the AVSSs of
fluctuation noise for the subset of the ARs magnetic fluxes
$\left|m_{\Phi_{th}^{I}}\right|$, $th\subseteq\\{350,700,900\\}$,
$I\subseteq\\{\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$, range
within [$1.2\pm 0.11$, $2.0\pm 0.12$], while approach 2 (Fig. 4, d) yields the
AVSSs for the same subset within [$1.0\pm 0.14$, $2.0\pm 0.16$]. The AVSSs for
the fluctuation spectra of magnetic flux of both polarities systematically
increase with the applied magnetic field threshold $th$ in this AR (see Table
3 and 3 rows 4,5,6). Accordingly, for $th=350$ G, the spectrum is close to the
$1/f-$ type one, with ranging within
$\left[\left|m_{\Phi_{350}^{I}}\right|\right]$ = [$1.0\pm 0.14$, $1.6\pm
0.12$], $I\subseteq\\{\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$
(black marks). For $th=700$ G and $th=900$ G, the well manifested pink noise
spectra inclined towards the red noise are seen, with
$\left[\left|m_{\Phi_{700}^{I}}\right|\right]$ = [$1.2\pm 0.15$, $2.0\pm
0.12$] (blue marks), and $\left[\left|m_{\Phi_{900}^{I}}\right|\right]$ =
[$1.4\pm 0.15$, $2.0\pm 0.16$] (red marks),
$I\subseteq\\{\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$,
respectively.

(vi) The situation is similar in the cases of the signed and unsigned magnetic
flux fluctuation spectra. In particular, approach 1 (Fig. 4, e) reveals the
AVSSs for the fluctuation noise of the signed and unsigned magnetic flux
$\left|m_{\Phi_{th}^{I}}\right|$, $th\subseteq\\{350,700,900\\}$,
$I\subseteq\\{\mbox{TSMF},\mbox{MSMF},\mbox{TUMF},\mbox{MUMF}\\}$, fall in the
range within [$1.1\pm 0.12$, $1.9\pm 0.11$]; while with approach 2 (Fig. 4,
f), we has the AVSSs range of [$0.8\pm 0.16$, $1.9\pm 0.15$]. Again, the AVSSs
for the signed and unsigned magnetic flux fluctuations systematically increase
with the applied magnetic field threshold $th$ (see Table 3 and 3 rows 7,8,9).
In particular, for $th=350$ G, the spectrum is close to the $1/f-$ type one,
with the AVSSs for unified subset of both approaches ranging within
$\left[\left|m_{\Phi_{350}^{I}}\right|\right]$ = [$0.8\pm 0.16$, $1.4\pm
0.13$], $I\subseteq\\{\mbox{TSMF},\mbox{MSMF},\mbox{TUMF},\mbox{MUMF}\\}$
(black marks). For $th=700$ G and $th=900$ G, the well manifested pink noise
spectra inclined towards the red noise are seen, with
$\left[\left|m_{\Phi_{700}^{I}}\right|\right]$ = [$1.2\pm 0.15$, $1.8\pm
0.15$] (blue marks), and $\left[\left|m_{\Phi_{900}^{I}}\right|\right]$ =
[$1.4\pm 0.15$, $1.9\pm 0.15$] (red marks),
$I\subseteq\\{\mbox{TSMF},\mbox{MSMF},\mbox{TUMF},\mbox{MUMF}\\}$,
respectively.

###  A.2 AR 12378

This is the AR with the magnetic field dispersed in many small sized
concentration areas, where the positive polarity areas are dominantly leading
and the negative polarity ones are trailing. The following features of the AR
12378 fluctuation noise spectra can be distinguished.

(i) Contrary to the case of AR 12381, the AVSSs of the AR 12378 area
fluctuations for the positive, negative, and total magnetic field,
$\left|m_{\Sigma_{th}^{J}}\right|$, $J\subseteq\\{+,-,tot\\}$, exhibit an
arbitrary behaviour, without any systematic relation to each other (Fig. 5 , a
-b, shown with crosses, pluses, and asterisks, respectively).

(ii) The spectral analysis in approach 1 (Fig. 5, a) shows that the AVSSs for
the full set of AR areas, $\left|m_{\Sigma_{th}^{J}}\right|$,
$th\subseteq\\{350,700,900\\}$, $J\subseteq\\{+,-,tot\\}$, fluctuation noise
range within [$1.1\pm 0.12$, $1.5\pm 0.12$], while approach 2 (Fig. 5, b)
yields the AVSSs within [$0.9\pm 0.16$, $1.5\pm 0.14$]. Contrary to the
compact AR 12381, the AVSSs for the area fluctuations of the AR 12378 do not
show any systematic monotonic dependence on the applied magnetic field
threshold $th$, but they are rather arbitrarily distributed (see Table 3 and 3
rows 1, 2, and 3). However, for $th=700$ G, certain systematic interrelation
is seen between the AVSSs of the area fluctuations obtained in the cases of
the positive $\left|m_{\Sigma_{700}^{+}}\right|$ (Fig. 5, a-b, blue crosses)
and negative $\left|m_{\Sigma_{700}^{-}}\right|$ (Fig. 5, a-b, blue pluses)
magnetic field, specifically, the first ones are always larger than the
latter. Moreover, for $th=350$ G and $th=700$ G, the spectra are close to the
$1/f-$ type ones, with ranging within
$\left[\left|m_{\Sigma_{350}^{J}}\right|\right]$ = [$0.9\pm 0.16$, $1.3\pm
0.12$] (black marks) and $\left[\left|m_{\Sigma_{700}^{J}}\right|\right]$ =
[$0.9\pm 0.15$, $1.3\pm 0.16$] (blue marks), $J\subseteq\\{+,-,tot\\}$,
respectively. Application of the threshold $th=900$ G yields a well manifested
pink noise spectrum with the AVSSs of subset ranging within
$\left[\left|m_{\Sigma_{900}^{J}}\right|\right]$ = [$1.1\pm 0.16$, $1.5\pm
0.14$], $J\subseteq\\{+,-,tot\\}$ (red marks).

(iii) Unlike the case of AR 12381, the AVSSs for the positive (leading sunspot
group) and negative (trailing sunspot group) total, namely,
$\left|m_{\Phi_{th}^{I}}\right|$, $I\subseteq\\{\mbox{TPMF},\mbox{TNMF}\\}$
(diamonds and triangles), and mean, namely, $\left|m_{\Phi_{th}^{I}}\right|$,
$I\subseteq\\{\mbox{MPMF},\mbox{MNMF}\\}$ (squares and circles) magnetic
fluxes in this AR do not show any monotonic interrelation, but are rather
arbitrarily distributed in various constructed spectra (Fig. 5, c-d).

(iv) Similarly to the case of different polarities, the AVSSs of both the
signed and unsigned total, namely, $\left|m_{\Phi_{th}^{I}}\right|$,
$I\subseteq\\{\mbox{TSMF},\mbox{TUMF}\\}$ (diamonds and triangles) and mean,
that is, $\left|m_{\Phi_{th}^{I}}\right|$,
$I\subseteq\\{\mbox{MSMF},\mbox{MUMF}\\}$ (squares and circles) also have
irregularly distributed values in the spectra (Fig. 5, e-f).

(v) The spectral analysis by approach 1 (Fig. 5, c) shows that the AVSSs of
fluctuation noise for the subset of the ARs magnetic fluxes
$\left|m_{\Phi_{th}^{I}}\right|$, $th\subseteq\\{350,700,900\\}$,
$I\subseteq\\{\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$, range
within [$1.1\pm 0.11$, $1.5\pm 0.12$]; while approach 2 (Fig. 5, d) yields the
AVSSs within [$1.0\pm 0.16$, $1.5\pm 0.14$]. The AVSSs for the fluctuation
spectra of magnetic flux of both polarities somewhat increase with the applied
magnetic field threshold $th$ in this AR (see Table 3 and 3 rows 4, 5, and 6).
However, the ranges of their values are significantly narrower than in the
case of AR 12381. Moreover, the spectra for all applied thresholds are close
to the $1/f-$ type one, with the AVSSs for unified subset for both approaches
of ranging within $\left[\left|m_{\Phi_{350}^{I}}\right|\right]$ = [$1.0\pm
0.16$, $1.3\pm 0.12$] (black marks),
$\left[\left|m_{\Phi_{700}^{I}}\right|\right]$ = [$1.0\pm 0.15$, $1.5\pm
0.12$] (blue marks), and $\left[\left|m_{\Phi_{900}^{I}}\right|\right]$ =
[$1.1\pm 0.16$, $1.5\pm 0.14$] (red marks),
$I\subseteq\\{\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$,
respectively.

(vi) The situation is similar in the cases of signed and unsigned magnetic
flux fluctuation spectra. In particular, approach 1 (Fig. 5, e) reveals the
AVSSs for the fluctuation noise of the signed and unsigned magnetic flux
$\left|m_{\Phi_{th}^{I}}\right|$, $th\subseteq\\{350,700,900\\}$,
$I\subseteq\\{\mbox{TSMF},\mbox{MSMF},\mbox{TUMF},\mbox{MUMF}\\}$, fall in the
range within [$1.1\pm 0.12$, $1.6\pm 0.11$]; while approach 2 (Fig. 5, f) has
the AVSSs range of [$0.9\pm 0.17$, $1.6\pm 0.15$]. Again, the AVSSs for the
signed and unsigned magnetic flux fluctuations somewhat increase with the
applied magnetic field threshold, $th,$ (see Table 3 and 3 rows 7, 8, and 9).
However, the ranges of their values are significantly narrower than in case of
AR 12381. Moreover, the spectra for all the thresholds are close to the $1/f-$
type one, with the AVSSs for unified subset of both approaches ranging within
$\left[\left|m_{\Phi_{350}^{I}}\right|\right]$ = [$0.9\pm 0.17$, $1.3\pm
0.13$] (black marks), $\left[\left|m_{\Phi_{700}^{I}}\right|\right]$ =
[$1.0\pm 0.15$, $1.3\pm 0.16$] (blue marks), and
$\left[\left|m_{\Phi_{900}^{I}}\right|\right]$ = [$1.2\pm 0.16$, $1.6\pm
0.16$] (red marks),
$I\subseteq\\{\mbox{TSMF},\mbox{MSMF},\mbox{TUMF},\mbox{MUMF}\\}$,
respectively.

###  A.3 AR 12524

This is an AR with a mixed internal structure, having two relatively large
spots with opposite polarities (similar to the ones constituting the compact
AR 12381) surrounded by many small areas of magnetic field concentration
predominantly of the same polarity that are similar to the case of the
dispersed AR 12378. The negative polarity areas are dominantly leading and the
positive polarity are the trailing ones. The following features of the AR
12524 fluctuation noise spectra can be distinguished.

(i) Again, contrary to the case of AR 12381 (and similarly to the case of AR
12378), the AVSSs of the AR 12524 area fluctuations for the positive,
negative, and total magnetic field, $\left|m_{\Sigma_{th}^{J}}\right|$,
$J\subseteq\\{+,-,tot\\}$, exhibit arbitrary values that do not reveal any
systematic relation between them (Fig. 6, a-b; shown as crosses, pluses, and
asterisks, respectively).

(ii) The spectral analysis via approach 1 (Fig. 6, a) shows that the AVSSs for
the full set of AR areas, $\left|m_{\Sigma_{th}^{J}}\right|$,
$th\subseteq\\{350,700,900\\}$, $J\subseteq\\{+,-,tot\\}$, fluctuation noise
range within [$1.1\pm 0.11$, $1.6\pm 0.11$], while approach 2 (Fig. 6, b)
yields the AVSSs within [$1.0\pm 0.15$, $1.5\pm 0.16$]. The picture of the
area fluctuation spectra in the case of this mixed type AR has partially the
properties of both types of the ARs considered above; in particular, this
refers to the AVSSs for the area, namely, the
$\left|m_{\Sigma_{th}^{J}}\right|$, $th\subseteq\\{350,700,900\\}$, and
$J\subseteq\\{+,-,tot\\}$ fluctuations of the AR 12524 (in some of the
spectrum measurement methods used here) show a certain monotonic increase,
depending on the applied increasing magnetic field threshold, $th,$ (similar
to the compact AR 12381), whereas this dependence is absent in the rest of the
used methods (similar to the dispersed AR 12378, see Table 3 and 3 rows 1, 2,
and 3). At the same time, for $th=350$ G and $th=700$ G, the spectra are close
to the $1/f-$ type ones with the AVSSs of subset ranging within
$\left[\left|m_{\Sigma_{350}^{J}}\right|\right]$ = [$1.0\pm 0.15$, $1.3\pm
0.16$] (black marks) and $\left[\left|m_{\Sigma_{700}^{J}}\right|\right]$ =
[$1.1\pm 0.15$, $1.5\pm 0.14$] (blue marks), respectively, while the well
manifested pink noise spectrum with the AVSSs of subset within
$\left[\left|m_{\Sigma_{900}^{J}}\right|\right]$ = [$1.2\pm 0.16$, $1.6\pm
0.11$] (red marks), $J\subseteq\\{+,-,tot\\}$.

(iii) In this AR, the AVSSs for the negative (leading sunspot group) and
positive (trailing sunspot group) total, namely,
$\left|m_{\Phi_{th}^{I}}\right|$, $I\subseteq\\{\mbox{TNMF},\mbox{TPMF}\\}$
(triangles and diamonds), and mean, namely, $\left|m_{\Phi_{th}^{I}}\right|$,
$I\subseteq\\{\mbox{MNMF},\mbox{MPMF}\\}$ (circles and squares) magnetic
fluxes do not show any monotonic interrelation similar to the case of the
dispersed AR 12378 (Fig. 6, c-d).

(iv) Similarly to the case of different polarities, the AVSSs of both the
signed and unsigned total, namely, $\left|m_{\Phi_{th}^{I}}\right|$,
$I\subseteq\\{\mbox{TSMF},\mbox{TUMF}\\}$ (diamonds and triangles) and mean,
i.e. $\left|m_{\Phi_{th}^{I}}\right|$,
$I\subseteq\\{\mbox{MSMF},\mbox{MUMF}\\}$ (squares and circles) also have
irregularly distributed values in the spectra (Fig. 6, e-f).

(v) The spectral analysis by approach 1 (Fig. 6, c) shows that the AVSSs of
fluctuation noise for the subset of the ARs magnetic fluxes,
$\left|m_{\Phi_{th}^{I}}\right|$, $th\subseteq\\{350,700,900\\}$,
$I\subseteq\\{\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$, range
within [$1.1\pm 0.11$, $1.8\pm 0.12$], while approach 2 (Fig. 6 (d)) yields
the AVSSs wihin [$1.0\pm 0.15$, $1.6\pm 0.16$]. The AVSSs for the fluctuation
spectra of the magnetic flux of both polarities systematically increase with
the applied magnetic field threshold $th$ as in the compact AR 12381 (see
Table 3 and 3 rows 4, 5, 6). Accordingly, for $th=350$ G, the spectrum is
close to the $1/f-$ type one, with the AVSSs for unified subset for both
approaches of ranging within $\left[\left|m_{\Phi_{350}^{I}}\right|\right]$ =
[$1.0\pm 0.15$, $1.3\pm 0.15$],
$I\subseteq\\{\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$ (black
marks). For $th=700$ G, the well manifested pink noise spectrum with the AVSSs
within $\left[\left|m_{\Phi_{700}^{I}}\right|\right]$ = [$1.1\pm 0.15$,
$1.6\pm 0.11$] (blue marks) is observed, which transforms for $th=900$ G to
the pink inclined towards the red noise spectrum with the AVSSs within
$\left[\left|m_{\Phi_{900}^{I}}\right|\right]$ = [$1.2\pm 0.17$, $1.8\pm
0.12$] (red marks),
$I\subseteq\\{\mbox{TPMF},\mbox{MPMF},\mbox{TNMF},\mbox{MNMF}\\}$.

(vi) The situation is similar in the cases of the signed and unsigned magnetic
flux fluctuation spectra. In particular, approach 1 (Fig. 6 (e)) reveals the
AVSSs for the fluctuation noise of the signed and unsigned magnetic flux
$\left|m_{\Phi_{th}^{I}}\right|$, $th\subseteq\\{350,700,900\\}$,
$I\subseteq\\{\mbox{TSMF},\mbox{MSMF},\mbox{TUMF},\mbox{MUMF}\\}$, fall in the
range within [$1.1\pm 0.11$, $1.8\pm 0.12$], while approach 2 (Fig. 6 (f))
gives the AVSSs within [$0.9\pm 0.16$, $1.7\pm 0.16$]. Again, the AVSSs for
signed and unsigned magnetic flux fluctuations depend on the applied magnetic
field threshold $th$ stronger than those in the case of the dispersed AR 12378
(see Table 3 and 3 rows 7,8,9). Furthermore, for $th=350$ G, the spectrum is
close to the $1/f-$ type one with the AVSSs for unified subset of both
approaches ranging within $\left[\left|m_{\Phi_{350}^{I}}\right|\right]$ =
[$0.9\pm 0.16$, $1.3\pm 0.16$],
$I\subseteq\\{\mbox{TSMF},\mbox{MSMF},\mbox{TUMF},\mbox{MUMF}\\}$ (black
marks). For $th=700$ G, the well manifested pink noise spectrum with the AVSSs
within $\left[\left|m_{\Phi_{700}^{I}}\right|\right]$ = [$1.2\pm 0.16$,
$1.5\pm 0.12$] (blue marks) is observed, which again transforms for $th=900$ G
to the pink inclined towards the red noise spectrum with the AVSSs within
$\left[\left|m_{\Phi_{900}^{I}}\right|\right]$ = [$1.2\pm 0.15$, $1.8\pm
0.12$] (red marks),
$I\subseteq\\{\mbox{TSMF},\mbox{MSMF},\mbox{TUMF},\mbox{MUMF}\\}$.

###  A.4 AR 12435 and AR 12437

These two unipolar active regions both have a dispersed internal structure
(similar to AR 12378). The AR 12435 has a positive polarity and appears in a
leading position in the considered sunspot group, whereas the AR 12437 has a
negative polarity and is in the trailing position. The following features of
the fluctuation noise spectra in the complex of AR 12435 and AR 12437 can be
distinguished.

(i) The AVSSs of the areas, $\left|m_{\Sigma_{th}^{+,-}}\right|$, fluctuations
of these two ARs interrelate to each other similarly to those of the dispersed
AR 12378 (Fig. 7, a-b), with crosses and pluses for AR 12435 and AR 12437,
respectively.

(ii) The spectral analysis via approach 1 (Fig. 7, a) shows that the AVSSs for
the full set of AR areas, fluctuation noise for AR 12435,
$\left|m_{\Sigma_{th}^{+}}\right|$, $th\subseteq\\{350,700,900\\}$, and AR
12437, $\left|m_{\Sigma_{th}^{-}}\right|$, $th\subseteq\\{350,700,900\\}$,
range within [$0.8\pm 0.13$, $1.2\pm 0.12$] and [$0.9\pm 0.12$, $1.5\pm
0.11$], respectively. Approach 2 (Fig. 7, b) yields the AVSSs within [$0.7\pm
0.17$, $1.2\pm 0.17$] and [$0.8\pm 0.17$, $1.4\pm 0.18$] for the AR 12435 and
AR 12437, respectively. Similarly to the dispersed AR 12378, the AVSSs for the
area $\left|m_{\Sigma_{th}^{+,-}}\right|$, fluctuations of both considered
unipolar ARs do not show any systematic dependence on the applied magnetic
field thresholds $th$ (see Table 3 and 3 rows 1, 2, and 3). The only
systematic interrelation seen, is that the AVSSs for the area
$\left|m_{\Sigma_{th}^{-}}\right|$, fluctuations of AR 12437 (blue pluses) are
always higher than those of $\left|m_{\Sigma_{th}^{+}}\right|$ of the AR 12435
(blue crosses). Furthermore, for all applied magnetic field thresholds $th$,
the spectra are close to $1/f-$ type for both ARs with the AVSSs ranging
within:

$\displaystyle\left[\left|m_{\Sigma_{350}^{+}}\right|\right]=[0.7\pm
0.17,1.1\pm 0.17]\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ $
$\displaystyle\left[\left|m_{\Sigma_{350}^{-}}\right|\right]=[0.8\pm
0.17,1.0\pm 0.12]\leavevmode\nobreak\ \text{(black marks);}$
$\displaystyle\left[\left|m_{\Sigma_{700}^{+}}\right|\right]=[1.0\pm
0.16,1.1\pm 0.16]\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ $
$\displaystyle\left[\left|m_{\Sigma_{700}^{-}}\right|\right]=[1.1\pm
0.15,1.4\pm 0.17]\leavevmode\nobreak\ \text{(blue marks);}$
$\displaystyle\left[\left|m_{\Sigma_{900}^{+}}\right|\right]=[1.0\pm
0.16,1.2\pm 0.17]\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ $
$\displaystyle\left[\left|m_{\Sigma_{700}^{-}}\right|\right]=[0.9\pm
0.16,1.5\pm 0.11]\leavevmode\nobreak\ \text{(red marks)}$

for the AR 12435 and AR 12437, respectively.

(iii) Similarly to the dispersed AR 12378, the AVSSs for the positive (AR
12435) and negative (AR 12437) total, namely,
$\left|m_{\Phi_{th}^{I}}\right|$, $I\subseteq\\{\mbox{TPMF},\mbox{TNMF}\\}$
(diamonds and triangles), and mean, i.e. $\left|m_{\Phi_{th}^{I}}\right|$,
$I\subseteq\\{\mbox{MPMF},\mbox{MNMF}\\}$ (squares and circles) magnetic flux
do not show any monotonic interrelation, but are rather arbitrarily
distributed in various constructed spectra (Fig. 7, c-d).

(iv) The analysis of the signed and unsigned magnetic fluxes is irrelevant in
this case of two unipolar ARs, whereas the case of total and mean magnetic
fluxes is addressed above in point (iii).

(v) The spectral analysis by approach 1 (Fig. 7, c) shows that the AVSSs of
fluctuation noise for the subset of the ARs magnetic fluxes
$\left|m_{\Phi_{th}^{I}}\right|$, $th\subseteq\\{350,700,900\\}$, range within
[$0.9\pm 0.13$, $1.3\pm 0.11$] and [$0.9\pm 0.13$, $1.5\pm 0.11$], for the AR
12435 ($I\subseteq\\{\mbox{TPMF},\mbox{MPMF}\\}$) and AR 12437
($I\subseteq\\{\mbox{TNMF},\mbox{MNMF}\\}$), respectively. Accordingly,
approach 2 (Fig. 7 (d)) yields the AVSSs within [$0.7\pm 0.16$, $1.2\pm 0.18$]
and [$0.8\pm 0.16$, $1.4\pm 0.17$], for the AR 12435 and AR 12437,
respectively. As in the case of the dispersed AR 12378, the AVSSs for the
fluctuation spectra of magnetic flux of both ARs slightly increase with the
applied magnetic field thresholds, $th$, and their ranges are narrower, as
compared to those of the compact AR 12381 (see Table 3 and 3 rows 4, 5, and
6). Furthermore, the spectra obtained for all $th$ are close to the $1/f-$
type spectrum, which is evidenced by the corresponding AVSSs ranges:

$\displaystyle\left[\left|m_{\Phi_{350}^{TPMF,MPMF}}\right|\right]=[0.7\pm
0.16,1.2\pm 0.16]\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ $
$\displaystyle\left[\left|m_{\Phi_{350}^{TNMF,MNMF}}\right|\right]=[0.8\pm
0.16,1.1\pm 0.12]\leavevmode\nobreak\ \text{(black marks);}$
$\displaystyle\left[\left|m_{\Phi_{700}^{TPMF,MPMF}}\right|\right]=[1.0\pm
0.17,1.2\pm 0.12]\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ $
$\displaystyle\left[\left|m_{\Phi_{700}^{TNMF,MNMF}}\right|\right]=[1.1\pm
0.15,1.4\pm 0.16]\leavevmode\nobreak\ \text{(blue marks);}$
$\displaystyle\left[\left|m_{\Phi_{900}^{TPMF,MPMF}}\right|\right]=[1.1\pm
0.16,1.3\pm 0.11]\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ $
$\displaystyle\left[\left|m_{\Phi_{900}^{TNMF,MNMF}}\right|\right]=[0.9\pm
0.16,1.5\pm 0.11]\leavevmode\nobreak\ \text{(red marks);}$

for the AR 12435 and AR 12437, respectively.

(vi) The comparison of approaches 1 and 2 for the analysis of the signed and
unsigned magnetic fluxes is irrelevant in this case of two unipolar ARs.

Figure 4: AVSSs obtained for the compact AR 12381 with approach 1 (left
panels) and approach 2 (right panels). Panels (a) and (b) panels show the
AVSSs of the AR area, fluctuation spectra. Panels (c) and (d) present the
AVSSs of the positive and negative magnetic flux fluctuation spectra. Panels
(e) and (f) show the AVSSs of the signed and unsigned magnetic flux
fluctuation spectra. In panels (a) and (b), the markers ’$+$’ and ’$\times$’
indicate $\left|m_{\Sigma_{th}^{+}}\right|$ and
$\left|m_{\Sigma_{th}^{-}}\right|$, respectively; whereas marker ’$\ast$’
denotes $\left|m_{\Sigma_{th}^{tot}}\right|$ (i.e. the combination of both
polarities). In panels (c) and (d), the markers ’$\diamond$’ and ’$\square$’
indicate $\left|m_{\Sigma_{th}^{TPMF,MPMF}}\right|$, while the markers
’$\triangle$’ and ’$\circ$’ are used for
$\left|m_{\Sigma_{th}^{TNMF,MNMF}}\right|$. In panels (e) and (f), markers
’$\diamond$’ and ’$\square$’ show $\left|m_{\Sigma_{th}^{TSMF,MSMF}}\right|$,
while markers ’$\triangle$’ and ’$\circ$’ denote
$\left|m_{\Sigma_{th}^{TUMF,MUMF}}\right|$, $th\subseteq\\{350,700,900\\}$.
The colours of the markers correspond to the applied magnetic field
thresholds: $th=350\;$G (black); $th=700\;$G (blue); $th=900\;$G (red). We
note that for the sake of better visualisation, the markers in all plots are
manually slightly shifted along the abscissa to avoid their overlapping.

Figure 5:  Same as Fig. 4, but for the dispersed AR 12378.

Figure 6: Same as Fig. 4, but for the mixed-type AR 12524.

Figure 7: Same as panels (a), (b), (c), and (d) in Fig. 4, but for the set of
two unipolar ARs, AR 12435 and AR 12437, of different polarity. In panels (a)
and (b), the markers ’$\times$’ and ’$+$’ denote the
$\left|m_{\Sigma_{th}^{+,-}}\right|$ of the AR 12435 and AR 12437,
respectively. In panels (c) and (d), the markers ’$\diamond$’ and ’$\square$’
indicate the $\left|m_{\Phi_{th}^{I}}\right|$,
$I\subseteq\\{\mbox{TPMF},\mbox{MPMF}\\}$ for the AR 12435, whereas
’$\triangle$’ and ’$\circ$’ signify the $\left|m_{\Phi_{th}^{I}}\right|$,
$I\subseteq\\{\mbox{TNMF},\mbox{MNMF}\\}$ for the AR 12437.