# The Sun’s Magnetic Power Spectra Over Two Solar Cycles. I. Calibration
Between SDO/HMI And SOHO/MDI Magnetograms

Yukun Luo  School of Space and Environment, Beihang University, Beijing,
People’s Republic of China  Key Laboratory of Space Environment Monitoring and
Information Processing of MIIT, Beijing, People’s Republic of China  Jie Jiang
School of Space and Environment, Beihang University, Beijing, People’s
Republic of China  Key Laboratory of Space Environment Monitoring and
Information Processing of MIIT, Beijing, People’s Republic of China  Ruihui
Wang  School of Space and Environment, Beihang University, Beijing, People’s
Republic of China  Key Laboratory of Space Environment Monitoring and
Information Processing of MIIT, Beijing, People’s Republic of China

###### Abstract

The Sun’s magnetic field is strongly structured over a broad range of scales.
The magnetic spatial power spectral analysis provides a powerful tool to
understand the various scales of magnetic fields and their interaction with
plasma motion. We aim to investigate the power spectra using spherical
harmonic decomposition of high-resolution SOHO/MDI and SDO/HMI synoptic
magnetograms covering three consecutive solar cycle minima in a series of
papers. As the first of the series, we calibrate and analyze the power spectra
based on co-temporal SDO/HMI and SOHO/MDI data in this paper. For the first
time, we find that the calibration factor $r$ between SOHO/MDI and SDO/HMI
varies with the spatial scale $l$ of the magnetic field, where $l$ is the
degree of a spherical harmonics. The calibration factor satisfies
$r(l)=\sqrt{-0.021l^{0.64}+2}\quad(5<\mathrm{l}\leq 539)$. With the
calibration function, most contemporaneous SOHO/MDI and SDO/HMI magnetograms
show consistent power spectra from about 8 Mm to the global scales over about
3 orders of magnitudes. Moreover, magnetic power spectra from SOHO/MDI and
SDO/HMI maps show peaks/knees at $l\approx 120$ corresponding to the typical
supergranular scale (about 35 Mm) constrained from direct velocimetric
measurements. This study paves the way for investigating the solar-cycle
dependence of supergranulation and magnetic power spectra in subsequent
studies.

magnetic power spectrum - calibration - supergranulation

##  1 Introduction

The solar magnetic field ubiquitously distributed on the solar surface is
highly structured and multiscale (Solanki et al., 2006). Magnetic features
have spatial scales from the global axial dipole field, to active regions
(ARs), to the network and internetwork magnetic fields, and further to scales
smaller than can currently be resolved (de Wijn et al., 2009). The well-
organized magnetic features result from the direct and continuous interaction
of the magnetic fields with turbulent convective flows. For instance, the
supergranulation velocity field is regarded to be responsible for the web-like
structure of the photospheric magnetic fields, i.e., network magnetic fields
(Leighton et al., 1962; Simon & Leighton, 1964; Rincon & Rieutord, 2018).

The kinetic and magnetic spatial power spectra over a wide range of scales
provide an effective way to explore the physical processes at the origin of
different magnetic features and their interaction with plasma flows (Nakagawa
& Priest, 1973; Nakagawa & Levine, 1974; Petrovay, 2001). Many efforts have
been taken to analyze the velocity maps for the kinetic power spectra. Using
spherical harmonic decomposition, Hathaway et al. (2000, 2015) obtain
photospheric kinetic energy spectra covering the degree of spherical
harmonics, $l$, from $l$=1 to $l\sim$3000 based on the high-resolution full-
disk Dopplergrams of MDI onboard the SOHO and HMI onboard the SDO,
respectively. Both spectra show distinct peaks representing granulation of
about 1 Mm and supergranulation of about 35 Mm. Since supergranulation is
strongly linked to the structure of the magnetic field, Williams & Pesnell
(2011) compare supergranular characteristics between the minima of cycles
22/23 and 23/24 using the same data and method as Hathaway et al. (2000).
Using the same method, Williams et al. (2014) further analyze Doppler data
from contemporaneous HMI and MDI observations. They show that supergranulation
appears to be smaller within HMI data than is measured with MDI.

Being different from studies on kinetic power spectra covering a wide range of
scales, most previous attempts on magnetic power spectra cover narrower
spatial scales, especially one of the two ends of the resolved scales. The
constantly improving spatial resolution of magnetic field measurements leads
to magnetic spectra studies in the upper end of the spectra covering granular
and subgranular scales. Using Fourier decompositions of local magnetograms of
the quiet Sun, Abramenko et al. (2001); Goode et al. (2010); Stenflo (2012);
Danilovic et al. (2016); Abramenko & Kutsenko (2020) studied magnetic power
spectra aiming to understand properties of internetwork magnetic field and its
possible origin, especially its connection with the small-scale dynamo (Vögler
& Schüssler, 2007; Marschalkó et al., 2013). The lower end of the magnetic
power spectrum corresponds to the global-scale field. Its properties provide
insights into the understanding of the solar cycle and the behavior of the Sun
as a star (Vidotto, 2016; Vidotto et al., 2018). DeRosa et al. (2012); Pipin &
Kosovichev (2018) decomposed full-disk synoptic magnetograms from MDI using
spherical harmonic decomposition. Their spectra span from $l$=1 to $l\sim$100
to understand the solar cycle and large-scale dynamo (Karak et al., 2014;
Charbonneau, 2020). Most of the rest of studies of the magnetic power spectra
concentrate on the analysis of local magnetograms of ARs for the relationship
between flare productivity and the magnetic power index (Abramenko et al.,
2002; Abramenko & Yurchyshyn, 2010; Mandage & McAteer, 2016) or for the
variation of magnetic power spectra during ARs flux emergence (Kutsenko et
al., 2019).

The aforementioned efforts on magnetic power spectra indicate that there are
very few studies of magnetic power spectra emphasizing supergranulation so
far. One of the relevant studies is made by Katsukawa & Orozco Suárez (2012),
whose magnetic power spectra show weak peaks at the supergranular scales based
on Hinode SOT magnetograms between 2006 November and 2007 December. Given the
association between supergranulation and the magnetic network, it is natural
to wonder about the following questions. What is the reliable magnetic power
spectrum in the broad spatial range covering the supergranular scale? Do the
spectra show peaks or knees at the supergranular scale? How do the power
spectrum and the scale corresponding to supergranulation vary within a solar
cycle and over multiple solar cycles? To our knowledge, all of the questions
have not been directly investigated as of now. Many studies on the cycle-
dependence of the supergranular size have been carried out by other data sets,
e.g., Ca II K images (McIntosh et al., 2011) and Doppler velocity data (De
Rosa & Toomre, 2004), which give quite controversial results.

Thanks to successive high-resolution full-disk magnetic field measurements
from MDI and HMI since 1996 onwards, we have an opportunity to investigate the
magnetic power spectra over a broad range of scales from the global scale to
supergranular scale and to a few Mm over cycles 23 and 24. This period covers
three cycle minima including the notable deep and extended cycle 23/24 minimum
(Jiang et al., 2013). Moreover, cycle 24 is the weakest cycle over the past
100 yr. These provide a rare opportunity to deeply investigate the cycle
dependence and activity dependence of the magnetic power spectrum and its
supergranulation scale. However, MDI and HMI adopt different spectral lines
and have different spatial resolutions and data qualities (Liu et al., 2012).
So to investigate the property of the magnetic power spectrum over multiple
cycles using measurements from the two instruments, the calibration between
the co-temporal HMI and MDI magnetograms is the prerequisite.

Two major methods are widely used to compare and calibrate line-of-sight
magnetograms and synoptic maps from different instruments. They are the pixel-
by-pixel comparison (e.g., Liu et al., 2012; Riley et al., 2014; Pietarila et
al., 2013; Tran et al., 2005) and histogram methods (e.g., Riley et al., 2014;
Jones, 1992; Jones et al., 2000; Wenzler et al., 2006). The calibration factor
varies somewhat with center-to-limb distance and field strength. But neither
method can answer whether the calibration factor varies with the spatial
scales of the magnetic field. Virtanen & Mursula (2017) introduce a new method
for scaling the photospheric magnetic field in terms of the harmonic expansion
and investigate the scaling of the harmonic coefficients between some pairs of
data sets (without the pair of MDI and HMI). The method can give an
independent scaling factor for the different harmonic terms, corresponding to
different spatial scales. The spatial-scale dependence of the calibration
factor affects the configuration of the magnetic power spectrum. This inspires
us to explore the calibration between MDI and HMI data in terms of the
spherical harmonic decomposition as the first step to investigating the
properties of the magnetic power spectrum, in this study.

In summary, this paper commences a series of investigations into the Sun’s
magnetic power spectrum aiming to quantify the solar cycle dependence of the
properties of the magnetic power spectrum over the broadest range of scales so
far. The objective of the first paper is twofold: spatial-scale dependent
calibration between MDI and HMI synoptic magnetograms and identification of
the supergranular scale based on calibrated MDI and HMI co-temporal magnetic
power spectra. The second paper will study the variation of the magnetic power
spectra after the calibration during 1996-2020 covering cycles 23 and 24 with
three consecutive cycle minima. We will put emphasis on the temporal variation
of the supergranular size and the power indices for the inertial range between
AR scales and supergranular scales in the course of the solar cycle and over 2
cycles. In the third paper, we will focus on the temporal variation of the
spectra at the range that is larger than AR scales.

This paper is organized as follows. We describe MDI and HMI synoptic
magnetograms and the method of obtaining the magnetic power spectra by
spherical harmonic decomposition in Section 2. In Section 3, we present our
calibration of co-temporal MDI and HMI magnetograms. The identification and
characterization of the supergranulation pattern based on the calibrated
magnetic power spectra are also presented in this section. We summarize and
discuss the above results in Section 4.

##  2 data and analysis

###  2.1 Data

In our series of studies, we use two radial synoptic magnetogram data sets
covering the 23rd and 24th solar cycles. They are observed by Michelson
Doppler Imager (MDI) on board the Solar and Heliospheric Observatory (SOHO)
(Scherrer et al., 1995) and the Helioseismic and Magnetic Imager (HMI) on
board the Solar Dynamics Observatory (SDO) (Scherrer et al., 2012; Schou et
al., 2012), respectively. The two instruments utilize different spectral lines
with various resolutions: the Ni I 6768 Å (Fe I 6173 Å) (Norton et al., 2006)
is adopted as the spectral line for the MDI (HMI), with coverage of
$3600\times 1080\ (3600\times 1440)$ pixels. The data sets begin in Carrington
Rotation (CR) 1911 (July 1996) and end in CR 2225 (December 2019, data missing
for CRs 1938-1942 and CR 1945). In this paper, we focus on analyzing the
synoptic maps for the overlap period, i.e. from CR 2097 to CR 2104.

The initial synoptic maps lack data on the polar magnetic fields, which are
crucial for the decomposition of the spherical harmonic function. Therefore,
we use polar-corrected data (Sun et al., 2011; Sun, 2018). Specifically, they
interpolate and smooth the north (south) polar magnetic fields by using a
multi-year series of well-observed polar fields from each September (March).
This process enables synoptic maps with polar field correction to be obtained.
It is worth noting that the polar correction approach of HMI differs slightly
from that of MDI, and the implications of these differences are analyzed in
Section 3.2.

###  2.2 Spherical harmonic decomposition and power spectra

For the magnetic field $B(\theta,\phi)$, we can express it by the following
Eq. (1)

$B(\theta,\varphi)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}B_{lm}Y_{lm}(\theta,\varphi),$
(1)

where $\theta$ is the colatitude, $\phi$ is the longitude,
$Y_{l,m}(\theta,\varphi)$ is the spherical harmonic function with degree $l$
and azimuthal order $m$, and $B_{l,m}$ is the corresponding spherical harmonic
coefficient.

Figure 1: CR 2097 synoptic map as an example of the spherical harmonic
expansion. An orthographic projection of the map centered at the equator and
the longitude of 0∘ is adopted. (a) the original synoptic map; (b)-(d)
spherical harmonics reconstruction up to the maximum degree $l_{max}=30,120$,
and 539, respectively. The magnetic fields in Panels (a)-(d) are saturated at
60, 8, 25, and 60 G, respectively.

For each synoptic map, we use the “pyshtools” package in Python to perform a
spherical harmonic decomposition. The algorithm is proposed by Driscoll &
Healy (1994), who employ a grid suitable for data with different sampling
densities at different latitudes and introduce weight coefficients. For
example, the longitudinal distances of the pixel points near the equator are
larger in synoptic maps than those near the polar region. Therefore, the data
need to be weighted when performing the spherical harmonic decomposition. The
discretization formula for obtaining the spherical harmonic coefficient for a
grid of size $2n\times 2n$, introducing the weighting factor $a_{j}$, is given
by:

$B_{lm}=\frac{\sqrt{2\pi}}{2n}\sum_{j=0}^{2n-1}\sum_{k=0}^{2n-1}a_{j}B(\theta_{j},\phi_{k})Y_{lm}(\theta_{j},\phi_{k}),$
(2)

where $\theta_{j}=\pi j/2n$, $\phi_{k}=\pi k/n$, corresponding to spherical
coordinates for any point on the sphere, and the number of grid points in both
latitude and longitude is $2n$. The weight coefficient $a_{j}$ is given by the
following equation

$\displaystyle a_{j}=\frac{2\sqrt{2}}{n}\sin\left(\frac{\pi
j}{\mathrm{n}}\right)\sum_{l=0}^{\frac{n}{2}-1}\frac{1}{2l+1}\sin\left([2l+1]\frac{\pi
j}{n}\right),$ (3) $\displaystyle\mathrm{j}=0,……,\mathrm{n}-1.$

This algorithm requires the grid to be $n\times 2n$ or $n\times n$. Therefore,
as the first step in data processing, we transform the data resolution to
$2160\times 1080$ for MDI and $2880\times 1440$ for HMI by a simple linear
two-point interpolation algorithm. Afterward, we also convert data from an
equal sine-latitude distribution to an equal latitude distribution. At this
step, we first calculate the sine of latitude at equal intervals. Then the
same two-point interpolation algorithm is used to transform the data based on
sine-latitudes. After these processes, the data can be used for “pyshtools”.

Due to the instrumental resolution limitations and sampling theorem, we
decompose the data up to $l=539$ and $l=719$ for MDI and HMI synoptic maps,
respectively. In the subsequent analysis, we consider only spherical harmonic
decomposition results of $l<540$ for both data sets. As an example, Figure 1
presents the magnetic field components at different spatial scales after
decomposing the CR 2097 synoptic map, which is shown in Figure 2 (a). Figure 1
(a) is the spherical projection of the original synoptic map. Panels (b)-(d)
show maps reconstructed by spherical harmonics up to different maximum
degrees. Large-scale structures become progressively more refined from Panels
(b) to (d) with $l_{max}$ increasing from 30 to 539. These expected results
illustrate the algorithm’s validity for the spherical harmonic decomposition
of the magnetograms. We can therefore use the results to obtain the power
spectra.

Using the orthogonality between the spherical harmonic functions, Parseval’s
theorem in the Cartesian coordinate system can be extended to the spherical
geometry. That is,

$\int_{\Omega}B^{2}(\theta,\phi)d\Omega=\sum_{l=0}^{\infty}P(l),$ (4)

where $P$ is the power spectrum and it is related to the spherical harmonic
coefficients by

$P(l)=\sum_{m=-l}^{l}B_{lm}^{2}.$ (5)

It is clear from Eq. (5) that in the power spectrum, information about the
scale of azimuthal order $m$ is combined. In our work, the spectral power
represents the magnetic energy at the spatial scale corresponding to the
spherical harmonic degree $l$. To convert the degree $l$ to the actual spatial
scale $\lambda$, we can use the following equation:

$\lambda=\frac{2\pi R_{\odot}}{\sqrt{l(l+1)}},$ (6)

where the parameter $R_{\odot}$ is the solar radius.

##  3 results

###  3.1 Scale dependence of MDI & HMI data

Figure 2: Magnetic power spectra obtained by MDI and HMI synoptic magnetograms
for CR 2097 (left) and CR 2101 (right). (a)-(b) Synoptic maps, magnetic fields
are saturated at 53 G and 100 G, respectively. (c)-(d) Power spectra of the
HMI map (red), MDI map (blue), and calibrated HMI map (black). (e)-(f) Power
spectra for $l_{max}=20$. The blue dotted lines are the power spectra of MDI
maps multiplied by a factor of 0.8 on polar fields. (g)-(h) Results of
supergranule identification based on the spectrum from $l$=6 to $l$=250. The
location of the peak or knee is marked with vertical gray dotted lines.

After the above data processing steps, we obtain the magnetic power spectra of
eight CRs during the overlapping period. As examples, Figures 2 (a) and (b)
show the synoptic magnetograms for CR 2097 and CR 2101, respectively, while
the corresponding HMI and MDI magnetic power spectra are shown in Figures 2
(c) and (d). The same figures but for the remaining six CRs are shown in the
appendix.

Figures 2 (c) and (d) show that both magnetic field strength and spatial
scales influence the difference between HMI and MDI power spectra, which is
consistent with previous studies (Liu et al., 2012; Riley et al., 2014;
Pietarila et al., 2013; Virtanen & Mursula, 2017). We next investigate whether
the impact of these two factors is minimal enough to use a fixed calibration
factor or whether it is necessary to calibrate based on spatial scales.

We present a comparison of the MDI and HMI spectral power for data during the
overlapping period in Figure 3. The red line represents the calibration
results obtained by Liu et al. (2012). The calibration factor, $1.4$,
corresponds to the vertical intercept in the log-log plot. The plot shows that
the distribution of points is non-linear, with a significant deviation from
the red line. That is, the calibration factor varies enough to make a fixed
calibration factor inapplicable. Thus we need to develop a new calibration
method applicable to power spectra analysis.

Notably, black dots in Figure 3 (small-scale part of magnetic fields)
generally have smaller calibration factors and lower spectral power than grey
dots (large-scale part). This may suggest that the calibration factors are
smaller for weaker magnetic fields. However, this seems to contradict the
findings of Liu et al. (2012), which indicate that the calibration factors are
smaller for stronger magnetic fields. This discrepancy may arise from the
stronger influence of spatial scale on the calibration factor than that of
magnetic strength, with smaller spatial scales corresponding to smaller
factors. These results indicate scale dependency in the difference between MDI
and HMI magnetograms. Therefore, it is essential to consider the calibration
at various scales in our work.

###  3.2 Influence of polar magnetic field on the power spectra

Figure 3: Comparison of HMI and MDI spectral power. The horizontal and
vertical axes are the spectral power of HMI and MDI, respectively. The gray
and black dots correspond to spherical harmonical degrees $l<200$ and $l>200$
respectively. The red line represents the fixed calibration factor 1.4, i.e.
$B^{2}_{l-MDI}=1.96B^{2}_{l-HMI}$ (1.4 for $B_{l}$).

In the theory of spherical harmonic decomposition, the term $l=0$ represents
the average result of the global scale magnetic field. For the magnetic field,
the term should be zero theoretically. However, this coefficient is not zero
based on observed magnetograms, which may be introduced by noise or during the
construction of the synoptic magnetograms (Bertello et al., 2014). We will not
consider this term in the analysis that follows. The power spectra in this
paper all start from $l=1$.

In Section 2.1, we mention that Sun et al. (2011); Sun (2018) use similar
approaches to correct the polar magnetic field of the HMI and MDI synoptic
magnetograms. However, the approaches require the use of the previous year’s
data for interpolation. But HMI synoptic magnetograms for the overlap period
do not have the previous year’s data. So for this part, they use the data of
MDI instead. In addition, the polar fields are smoothed and weighted with the
non-polar data in the correction. These correction methods increase
uncertainties of the polar magnetic field and further influence the magnetic
power spectra. Therefore, we need to determine the calibration range according
to the range of the power spectra affected by the polar field.

We multiply the data above 75° north and south of MDI magnetograms by a factor
of 0.8 to test the effect of the polar field. We then obtain their power
spectra, which are shown in Figures 2 (e) and (f). It can be seen that the
scaled MDI spectral power corresponding to the dash line has a certain
decrease at $l=1,3,5$ and are close to HMI. This just supports the idea that
the polar magnetic field has an impact on the power spectra of the large scale
field ($l\leq 5$). The results are similar to Virtanen & Mursula (2016): The
correction of the polar field affects axial dipole ($l=1$) and quadrupole
($l=2$) but has no effect on the results of a smaller scale. Thus, the effect
of the polar field uncertainty for $l>5$ can be ignored. Taking these results
together, we will calibrate the power spectra for $l>5$ and the calibration
results will be applied in the non-overlap period.

###  3.3 Scale dependent calibration function for MDI and HMI magnetograms

Figure 4: Spatial scale dependency of the calibration factor between MDI and
HMI spectral power. Black and gray dots are the observation data within and
outside $\sigma$, the variance of the calibration factors for a given $l$,
respectively. The blue line is the mean of the black dots and is fitted to
give the scale-dependent calibration function $r(l)$ (red line). Table 1:
Quantitative analysis and comparison of calibration results

CR | Consistent interval of $l$a |  $RSS(r=1)$b |  $RSS(r=1.4)$b |  $RSS(r(l))$b |  $\Delta B_{r(l)}$/$\Delta B_{r=1.4}$
---|---|---|---|---|---
2097 | 6-251 | 42.23 | 32.07 | 2.26 | 0.806
2098 | 6-178 | 24.73 | 48.67 | 5.55 | 0.810
2099 | 6-539 | 374.25 | 87.21 | 2.84 | 0.822
2100 | 6-539 | 163.74 | 29.12 | 4.19 | 0.812
2101 | 6-539 | 893.03 | 121.02 | 4.32 | 0.785
2102 | 6-539 | 318.19 | 91.14 | 3.68 | 0.800
2103 | 6-539 | 169.69 | 75.51 | 1.63 | 0.819
2104 | 162-539 | 153.16 | 436.56 | 145.14 | 0.807

  * a

Consistent interval is the range for $l$ where MDI and calibrated HMI power
spectra are consistent.

  * b

Residual sum of squares between the MDI and HMI power spectrum calibrated with
the calibration factor $r$=1 (uncalibrated one), $r$=1.4, and calibration
function Eq.(7), respectively.

In the first two subsections, we have shown that the calibration factors are
scale-dependent in the range $l>5$. To investigate this further, we combine
data for the eight CRs with $l>5$ and explore the pattern of variation of the
spectral power ratio $B^{2}_{l-MDI}/B^{2}_{l-HMI}$ with the spherical harmonic
degree $l$. The result is shown in Figure 4. The plot clearly shows a
decreasing trend in the ratio as the degree $l$ increases and this trend is
consistent across different CRs. This further suggests that the calibration
factor is a function of $l$.

Due to the limited number of CRs with the same $l$, the impact of noise or
measurement error is relatively large. To mitigate the impact, we exclude
special points by discarding data larger than the variance $\sigma$ for a
given $l$. The remaining data points are then averaged to obtain the solid
blue line in Figure 4. We use a nonlinear function to fit the averaged data
and obtain the following calibration function $r(l)$:

$r(l)=\frac{B_{l-MDI}}{B_{l-HMI}}=\sqrt{-0.021\ell^{0.64}+2.02}.$ (7)

The calibration factor $r$ for $l=5\ (\approx 800$Mm) is 1.4, while for $l=30\
(\approx 140$Mm) and $l=140\ (\approx 30$Mm), the calibration factors are 1.35
and 1.23, respectively. The factors are close to the calibration factor of 1.4
obtained in Liu et al. (2012). These demonstrate that the calibration factors
obtained through traditional methods for analyzing full-disk magnetograms are
mostly relevant for large-scale structures, such as active regions. Our
results suggest that the classical methods may have limited applicability to
smaller-scale features. In contrast, Riley et al. (2014) found a calibration
factor $r\approx 1.05$ for the CR 2097 synoptic magnetogram. This value is
smaller than our calibration factor for the large-scale section but larger
than that for the small-scale section. As shown in Figure 2 (c), small-scale
features contribute significantly to the total power, which suggests that the
overall single calibration factor obtained by averaging our calibration
factors may be similar to the value of 1.05.

In addition, Virtanen & Mursula (2017) concluded that the calibration factors
for low spherical degrees are smaller than those obtained through pixel-by-
pixel comparison or histogram techniques when they performed spherical
harmonic decomposition for synoptic maps. However, it should be noted that
their work did not involve the calibration of HMI and MDI data sets, thus the
applicability of their conclusion to the calibration of these two data sets is
uncertain. Hence, we should be careful to directly compare our findings with
theirs.

###  3.4 Evaluation of the new calibration method

After applying the calibration function Eq. (7) to the power spectra, we first
evaluate the similarity between the calibrated HMI and MDI power spectra.
Specifically, we consider the calibrated HMI and MDI power spectra to be
consistent if the difference between them is less than $20\%$ of the MDI
spectral power. We list the consistent interval of $l$ based on this criterion
in the second column of Table 1. Our calibration method leads to throughout
consistent HMI and MDI power spectra for CRs 2099-2103, based on this
criterion. Moreover, the power spectra of CR 2097 and CR 2098 exhibit
consistency only on the large-scale part, from $l=6$ to around $l\approx 200$.
And the consistent interval for CR 2104 is mainly in the small-scale part.

The exception of CR 2104 is also apparent in the corresponding images. In
Figure 3, below the red line, there is a cluster of gray dots, which results
from CR 2104. In Figure 4 gray points below the black dots show a significant
deviation. This part of the data is also from CR 2104. These results suggest
that there might be some issues with the CR 2104 HMI or MDI magnetograms,
which could be attributed to instrumental measurements.

To illustrate in more detail the applicability of our calibration method for
the power spectra analysis, we additionally use two quantitative comparisons.
The residual sum of squares ($RSS$) measures the difference between the MDI
power spectra and the uncalibrated ($r=1$) or calibrated HMI power spectra.
The formula is as follows.

$RSS=\sum_{l=6}^{539}\left(B^{2}_{l-HMIc}-{B}^{2}_{l-MDI}\right)^{2}.$ (8)

The results are shown in columns 3 to 5 of Table 1, where $r=1.4$ represents
the fixed calibration factor 1.4 and $r(l)$ represents the calibration
function Eq. (7). The smaller the RSS value is, the smaller difference between
the two power spectra is. While using a fixed calibration factor, the RSS is
reduced to only one-fifth to one-half of the uncalibrated case. For CR 2098
and CR 2104, the RSS even shows an increase. However, when using the
calibration function $r(l)$, all the RSS show a decrease, with the vast
majority reduced to a tenth or even a hundredth of the uncalibrated case. This
shows that our calibration method is more suitable for power spectra analysis.

In addition to using the RSS to evaluate the effect of our calibration on the
power spectra, we also quantify the effect on the magnetograms. To accomplish
this, we perform an inverse spherical harmonic decomposition of the power
spectra to reproduce the magnetograms. Using the MDI magnetograms as the
reference, we separately obtain the HMI magnetogram using the two calibration
methods (fixed calibration factor 1.4 and calibration function $r(l)$). The
ratio of the difference between the HMI and MDI magnetograms at a given pixel
is obtained using the following formula:

$\frac{\Delta B_{r(l)}}{\Delta
B_{r=1.4}}=\frac{B_{MDI}-B_{HMI-r(l)}}{B_{MDI}-B_{HMI-r=1.4}}.$ (9)

For each CR map, we then conduct a linear regression of the results for all
pixels. The results are summarized in the last column of Table 1. Notably, the
ratios of magnetogram differences for all CRs are within a narrow range of
$0.81\pm 0.01$. These quantitative comparisons indicate the efficacy of our
calibration approaches in achieving not only a better analysis of power
spectra but also a more consistent between HMI and MDI synoptic maps.

###  3.5 Supergranule identification

Table 2: Location of peak or knee identified on power spectra of MDI and
calibrated HMI synoptic magnetograms for CRs 2097-2104.

CR | $l_{sgr1}(HMI)$ | $l_{sgr1}(MDI)$ | $l_{sgr2}(HMI)$ | $l_{sgr2}(MDI)$ | $l_{sgr3}(HMI)$ | $l_{sgr3}(MDI)$
---|---|---|---|---|---|---
2097 | 130 | 130 | 131±5 | 132±5 | 129 | 130
2098 |  |  |  |  | 138 | 133
2099 | 140 | 140 | 156±5 | 156±5 | 143 | 139
2100 | 123 | 123 | 128±5 | 128±5 | 127 | 127
2101 | 123 | 120 | 119±5 | 119±5 | 117 | 117
2102 |  |  |  |  | 133 | 132
2103 |  |  |  |  |  |
2104 |  |  |  |  | 125 | 125

  * •

Notes. The symbols ‘$l_{sgr1}$’, ‘$l_{sgr2}$’, and ‘$l_{sgr3}$’ are degrees of
spherical harmonics of the identified peak or knee corresponding to
supergranulation using the three methods presented in Section 3.5.

The analysis of Dopplergrams using spherical harmonic functions, as discussed
in Introduction, shows the peak around $l=120$ on the kinetic power spectra
corresponding to supergranulation. In Section 3.3 we have mentioned that MDI
and calibrated HMI power spectra are consistent around this degree. So it is
feasible to search for supergranule features in the magnetic power spectra.

Based on Figure 2 and the power spectra images in the appendix, it is apparent
that a majority of the magnetic power spectra exhibit peaks or knees around
$l=120$. Here we use three algorithms to verify the existence of these
structures and the corresponding specific spatial scales. We start with a
morphological approach based on the code developed by Duarte & Watanabe (2021)
to find the peaks in the spectra. We search and identify peaks by the trend of
the curve around the local maximum, the threshold, and the width. By adjusting
these parameters, the identification results can be optimised. The
corresponding results are in columns 2 and 3 ($l_{sgr1}$) of Table 2. The
second method is similar to wavelet analysis, where a wavelet is fitted to the
peak and the presence of a peak is discerned based on the result of fit (Du et
al., 2006). This method is achieved through the ‘scipy.signal.find_peaks_cwt’
within Python. Since the method does not give the peak, but gives the center
of the wavelet, we list the range of the peak rather than the exact value,
with the results in columns 4 and 5 ($l_{sgr2}$) of Table 2. The third method
we use is the algorithm for finding peaks and knees used in the field of
neural power spectra: FOOOF (Donoghue et al., 2020). This method gives
information on the peaks of the four CRs around the supergranule scale and
also gives the knee of the three CRs. The results are in columns 6 and 7
($l_{sgr3}$) of Table 2.

The supergranulation identification results are between $l=116\ (36$ Mm) and
$l=140\ (30$ Mm), which are consistent with the results in the kinetic power
spectra. Notably, for the same CR, we observe slight differences in the
results obtained from different methods. For instance, in the case of CR 2099,
the first two methods yield a difference of $\Delta l=16$. The third method,
meanwhile, can assist in determining which result is more reliable. Therefore,
by combining the three methods, we can identify the supergranule scale
correlated features of the power spectra of most CRs. However, we are unable
to identify significant features in the power spectra of CR 2103, possibly due
to the portion of power spectra belonging to active regions masking
supergranulation.

Furthermore, in the work of Williams et al. (2014), they observe different
sizes of supergranulation in the kinetic power spectra for HMI and MDI, with
$l=132$ for HMI and $l=119$ for MDI. They attribute the difference to the
different spatial resolutions. They show that by smoothing the resolution of
HMI Dopplergrams, a power spectrum similar to MDI data can be obtained.
However, our identifications do not indicate any significant differences
between the HMI and MDI results, which remain generally consistent.

Regarding the absence of differences between the supergranular scales in our
results mentioned above, we first exclude the effect of the calibration. We
note that the difference of calibration factors ($\Delta r=0.02$) around the
supergranule scale cannot significantly alter the scale of the feature.
Furthermore, the identified supergranule sizes in HMI and MDI maps of CR 2104
remain consistent, even though their power spectra do not coincide in $l<162$.
We suggest that the difference in the identification results is due to the
influence of small-scale structures. Hathaway et al. (2002); Rincon et al.
(2017) mention that the kinetic power spectra near the size of
supergranulation can be affected by granulation, which are not completely
removed. The higher the resolution, the more features of granulation are
included. Therefore, smoothing maps with high resolution is essentially a way
to obtain similar power spectra by reducing the influence of small-scale
structures. Magnetic power spectra in this paper are obtained from synoptic
magnetograms, which have a relatively low resolution. Multiple full-disk
magnetograms are averaged in the construction of a synoptic map. Strong small-
scale magnetic structures and noise are removed. Therefore, we can obtain
ideal results in the magnetic power spectra. On the other hand, in addition to
small-scale structures, large-scale structures such as active regions may also
cause an unphysical shift of the position of supergranulation in the power
spectra. We need to be careful about this potential effect in future work.

##  4 conclusion and discussion

In this work, we conduct spherical harmonic decomposition of HMI and MDI
synoptic magnetograms for their overlapping period to obtain magnetic power
spectra. A scale-dependent calibration function between MDI and HMI spectral
power $r(l)=\sqrt{-0.021\ell^{0.64}+2.02}$ for magnetic power spectra analysis
is derived. In some magnetic power spectra, the peak or knee around $l=120$
associated with supergranulation is clearly presented. This study lays the
foundation for further research on the cycle dependence of supergranulation.

The scale-dependent calibration factors might be mainly attributed to the
different effects of magnetic filling factors on the measurement of magnetic
fields with different scales. The magnetic filling factor is set to the
constant value of 1 when both the MDI and HMI magnetograms were derived from
polarimetric measurements (Berger & Lites, 2003; Liu et al., 2012; Centeno et
al., 2014; Griñón-Marín et al., 2021). As shown in Figure 1, the larger-scale
magnetic field is more uniformly distributed. For the smaller-scale field
having distinct magnetic structures, lower-resolution data tend to give lower
magnetic field strength since only the mean flux density contributes to the
Stokes profiles. Hence we see the scaling factor $r(l)$ decrease with $l$ when
we scale higher-resolution HMI data to lower-resolution MDI data. In addition,
the different modulation transfer functions for the two instruments could be
the other ingredient contributing to the scale-dependent calibration factors
(Abramenko et al., 2001; Stenflo, 2012), especially near the resolution limit
of the MDI maps, i.e., around the largest $l$. The scale-dependent calibration
factors have been illustrated by Virtanen & Mursula (2017), who investigated
different pairs of datasets. But note that they scaled lower-resolution data
to higher-resolution one, and the scaling factors typically increase with $l$.
There is no particular reason for us to choose the lower-resolution data, MDI
synoptic maps, as the reference in this paper.

In contrast to a constant calibration factor, our proposed calibration
function exhibits superior suitability for power spectra analysis while
simultaneously reducing the discrepancies between magnetograms after
calibration. Furthermore, as highlighted in Virtanen & Mursula (2017), the
calibration approach is independent of the resolution of magnetograms, thereby
reducing the potential impact of resolution on the calibration results.
Despite the limited number of synoptic magnetograms available during the
overlap period of MDI and HMI, our analysis of the power spectra of these
magnetic maps reveals a consistent and scale-dependent variation of the
calibration factors, with no apparent signs of temporal variation. We limit
our scale-dependent calibration between HMI and MDI synoptic magnetograms to
the range $l>5$ because of the unreliable measurement of the polar field. The
calibration function could be extended to the range $l\leq 5$, which gives $r$
about 1.4. The value is essential for the understanding of the polar field
distribution and long-term evolution of the solar open flux (Jiang et al.,
2013; Linker et al., 2017).

So far there are very few studies of supergranulation-scale magnetic fields
with power spectra of MDI or HMI synoptic maps. For the first time, to the
best of our knowledge, we derive the magnetic power spectrum over the broad
spatial scale from about 20 Mm to the global scale and identify
supergranulation based on the magnetic spectra. Although what we detected is
actually the magnetic network, rather than the supergranulation flow, the
magnetic network has a direct association with supergranulation (Rincon &
Rieutord, 2018).

Most magnetic power spectra we investigated show a distinct peak or knee at
the supergranular scale of about 35 Mm ($l\approx 120$), which is consistent
with the typical size measured from the Doppler velocity observations. The
consistency is expected given the association between supergranulation and the
magnetic network and the consistency also is a piece of evidence to support
the association. In contrast, some previous investigations on the scale of
supergranulation using different methods show divergent results. For example,
applying local correlation tracking techniques to Doppler velocity data of
MDI, De Rosa & Toomre (2004) derived the average supergranular cell diameter
lies in the 12–20 Mm range. Meunier et al. (2008) show the average size is in
the range of 16-17 Mm. However, the supergranulation scale is not necessarily
a distinct feature on all magnetic power spectra as presented in Table 2 since
active regions tend to disorganize and wash away supergranulation (Hindman et
al., 2009). Due to the effects of active regions and other sources of noise,
the identification of the peak or knee of the power spectra is slightly
method-dependent. The slightly different locations of the peak or knee
correspond to slightly different sizes of supergranulation. Besides this
random source for the variation of supergranular sizes, there may also exit an
inherent variation of supergranular sizes with solar activity (Meunier et al.,
2007).

Knowledge of the variation of the supergranular size presented by the magnetic
power spectrum is crucial for a better understanding of the interaction
between the magnetic field and convection at different scales and of the solar
total and spectral irradiance variations (Yeo et al., 2014). In our subsequent
work, we will refine our identification methods of supergranulation from
magnetic power spectra and extend our analysis to the entire solar cycles 23
and 24. We will investigate how the size of supergranulation varies within a
solar cycle and over multiple cycles.

We thank the referee for the valuable comments. The research is supported by
the National Natural Science Foundation of China No. 12173005 and National Key
R&D Program of China No. 2022YFF0503800. J.J. acknowledges the International
Space Science Institute Teams 474. We would like to express our gratitude to
the teams responsible for the development of Python toolkits such as
‘pyshtools’ and ‘scipy’. The SDO/HMI data are courtesy of NASA and the SDO/HMI
team. SOHO is a project of international cooperation between ESA and NASA.

Figure 5: Same as Figure 2, but for CR 2098 (left) and CR 2099 (right). Figure
6: Same as Figure 2, but for CR 2100 (left) and CR 2102 (right). Figure 7:
Same as Figure 2, but for CR 2103 (left) and CR 2104 (right).

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