11institutetext: Max-Planck Institut für Sonnensystemforschung, Justus-von-
Liebig-Weg 3, 37077 Göttingen, Germany
11email<EMAIL_ADDRESS>22institutetext: School of Space Research, Kyung Hee
University, Yongin, 446-701 Gyeonggi, Korea

# The variation in the response of solar full-disc magnetographs

K. L. Yeo 11 S. K. Solanki 1122 N. A. Krivova 11

###### Abstract

Context. The utility of full solar disc magnetograms as a long-term record of
the photospheric magnetic field requires an understanding of how stable these
observations are with time and the systematic differences between the various
instruments.

Aims. We compared magnetograms from the KPVT/SPM, SoHO/MDI, SOLIS/VSM, and
SDO/HMI with the aim of probing the effect on measured solar magnetism of the
variation in instrument response with time, magnetogram signal level, and
position on the solar disc.

Methods. Taking near-simultaneous observations from the various instruments,
we examined the surface coverage by magnetic activity and the effect of cross-
calibrating the various instruments under different assumptions.

Results. By comparing the surface coverage by magnetic activity in the
observations from the various instruments, we traced the effect of the time
variation in instrument response on the longitudinal magnetogram signal and
disc-integrated unsigned magnetic flux. This yielded evidence of acute changes
in the response of MDI and VSM with certain events such as the SoHO vacation
in 1998 and the upgrade of the VSM CCD camera in 2009. Excluding these
changes, the effect of instrument instability on the magnetogram signal and
disc-integrated magnetic flux appears to be rather benign, with an associated
uncertainty of less than 2$\%$. We determined the magnetogram signal ratio
between each instrument pairing as a function of magnetogram signal level and
distance from disc centre and with it cross-calibrated the various
instruments. We compared the result with that from repeating the cross-
calibration with the overall magnetogram signal ratio. This allowed us to
estimate the uncertainty in the magnetogram signal associated with the
variation in instrument response with magnetogram signal level and distance
from disc centre to be about 8$\%$ to 14$\%$. The corresponding uncertainty in
the disc-integrated magnetic flux is about 7$\%$ to 23 $\%$.

Conclusions. To the best of our knowledge, this is the first study of its kind
to quantify the uncertainty in measured magnetism from the variation in
instrument response with time, magnetogram signal level, and disc position.
The results here will be useful to the interpretation of SPM, MDI, VSM, and
HMI magnetograms. As examples, we applied our findings to selected results
from earlier studies based on such data.

###### Key Words.:

Sun: magnetic fields

##  1 Introduction

Since the 1960s, a succession of solar telescopes recorded longitudinal
magnetograms of the full solar disc at sub-minute to daily intervals. This
body of observations, giving the spatial distribution and time evolution of
the photospheric magnetic field over the past half-century, is of obvious
importance to the understanding of solar magnetism. Through its application to
models of space weather (Tóth et al., 2005; Bobra & Couvidat, 2015) and solar
irradiance variability (Krivova et al., 2003; Wenzler et al., 2004, 2006; Ball
et al., 2012; Yeo et al., 2014), which provides the solar forcing input
required by climate simulations (Haigh, 2007; Ermolli et al., 2013; Solanki et
al., 2013), this body of observations is also pertinent to the study of Sun-
Earth interactions.

Measured photospheric magnetic flux density, i.e. the magnetogram signal, is
modulated by instrumental factors such as the design, calibration, stray
light, spectral line surveyed, spatial and spectral resolution, and specifics
of the data reduction. The result is the systematic differences between the
observations from the various instruments. This is most readily apparent in
the conversion factor, the ratio of the magnetogram signal registered by two
given instruments, deviating from unity (see Fig. 9 in Riley et al., 2014).
Reconciling the observations from the various instruments is complicated by
the time variation in instrument response, which can arise from hardware
changes, thermal fluctuations, and ageing and exposure degradation.

Few studies have examined the stability of full-disc magnetographs. Arge et
al. (2002) compared synoptic magnetograms from the Mount Wilson Observatory,
Wilcox Solar Observatory, and Kitt Peak Vacuum Telescope (KPVT). The authors
looked at the total photospheric magnetic flux and open magnetic flux and
noted that while the various data sets are broadly consistent with one
another, there are certain discrepancies which they attributed to instrument
changes. Riley et al. (2014) extended this study by including similar
observations from the Global Oscillation Network Group (GONG, Harvey et al.,
1996), the Michelson Doppler Imager onboard the Solar and Heliospheric
Observatory (SoHO/MDI, Scherrer et al., 1995) and the Helioseismic and
Magnetic Imager onboard the Solar Dynamics Observatory (SDO/HMI, Scherrer et
al., 2012). The authors examined the conversion factor as a function of time
to trace the relative instrument response. Similar to Arge et al. (2002),
Riley et al. (2014) reported that while the conversion factor between the
various magnetographs appears to be broadly stable with time, there are
indications of instrument changes. While such studies have highlighted
instrument changes, the effect on measured magnetism, such as the observed
magnetic flux density and amount of magnetic flux, has not been investigated
in detail.

Besides Riley et al. (2014), a number of other studies examined the conversion
factor between the various full-disc magnetographs. The conversion factor was
found to vary with magnetogram signal level (Jones & Ceja, 2001; Wenzler et
al., 2004; Liu et al., 2012; Pietarila et al., 2013; Riley et al., 2014; Yeo
et al., 2014) and position on the solar disc (Tran et al., 2005; Demidov et
al., 2008; Pietarila et al., 2013; Riley et al., 2014; Yeo et al., 2014).
Ideally, the magnetogram signal at a given point should scale linearly with
the actual magnetic flux density there and be independent of the location of
the point on the solar disc. The variation in the conversion factor with
magnetic flux density and disc position suggest however, that instrument
response fluctuates such that the magnetogram signal at a given point can
scale nonlinearly with the actual magnetic flux density there and change with
the location of the point on the solar disc. This can arise from
inhomogeneities in instrumental properties across the image plane and
uncertainties in the calibration and data reduction. As with the time
variation in instrument response, the effect of the changes in instrument
response with magnetogram signal level and disc position on measured magnetism
has not been investigated in detail.

The variation in instrument response with time, magnetogram signal level and
disc position is relevant to the reconciliation of the observations from the
various full-disc magnetographs and in consequence the utility of this body of
observations as a long-term record of photospheric magnetism. In this study,
we aim to address the issue that the effect of such fluctuations in instrument
response on measured magnetism is not well studied. For this purpose, we
compare longitudinal magnetograms from the higher spatial resolution (pixel
scale of 2 arcsec and finer) full-disc magnetographs that had operated over
the past three decades (Table 1): the KPVT SPectroMagnetograph (SPM, Jones et
al., 1992), SoHO/MDI, the Vector SpectroMagnetograph in the Synoptic Long-term
Investigations of the Sun instrument suite (SOLIS/VSM, Keller et al., 2003),
and SDO/HMI. From an examination of the solar disc coverage by magnetic
features and the conversion factor between the various instruments, we will
estimate the uncertainty in the observed magnetic flux density and disc-
integrated photospheric magnetic flux associated with the fluctuations in
instrument response. In the following, we will describe the data selection and
reduction (Sect. 2) and the aforementioned analysis (Sect. 3), before a
discussion of the results (Sect. 4) and a summary of the study (Sect. 5).

Table 1: Overview of the instruments examined in this study.

|  |  | Image size | Pixel scaleaa$a$The nominal spatial resolution is, for each instrument, about twice the pixel scale, though for ground-based telescopes such as SPM and VSM, this spatial resolution is never reached due to atmospheric seeing.
---|---|---|---|---
Instrument | Period [year.month.day] | Spectral line | [pixel] | [arcsec]
SPM | 1992.11.19 - 2003.09.21 | Fe I 8688 Å | $1788\times 1788$ | 1.14
MDI | 1996.03.19 - 2011.04.11 | Ni I 6768 Å | $1024\times 1024$ | 1.98
VSMbb$b$The CCD camera of VSM was upgraded in the gap between 2009.11.27 and 2009.12.18, giving the change in pixel scale. | 2003.08.23 - 2009.11.27 | Fe I 6302 Å | $2048\times 2048$ | 1.125
| 2009.12.18 - 2017.10.27 |  |  | 1
HMI | 2010.04.30 - present | Fe I 6173 Å | $4096\times 4096$ | 0.504
111

##  2 Data selection and reduction

In this study, we compare full-disc longitudinal magnetograms from KPVT/SPM
(Jones et al., 1992), SoHO/MDI (fd_M_96m_lev182 data series, Scherrer et al.,
1995), SOLIS/VSM (Keller et al., 2003), and SDO/HMI (M_720s data series, Schou
et al., 2012). In Table 1, we list the period of operation, spectral line
surveyed, image size, and pixel scale of each instrument.

###  2.1 Exceptional instrument events

First, let us describe certain events that occurred to MDI and VSM, which are
exceptional in that they are outside the normal operation of such instruments,
that altered their spatial resolution (c.f., Sects. 2.3 and 3.1). We exclude
these events from the assessment of instrument stability by considering MDI
and VSM observations from before and after each event separately.

The SoHO spacecraft experienced a series of extended outages between June 1998
and February 1999, often referred to as the SoHO vacation, during which MDI
might have degraded (see discussion in Ball et al., 2012). Also, the
instrument was refocused after the SoHO vacation. Let us denote MDI before and
after the SoHO vacation and the refocus of this instrument as MDI(1) and
MDI(2), respectively.

The CCD camera of VSM was upgraded in December 2009, changing the pixel scale
from 1.125 arcsec to 1 arcsec (Table 1). The instrument team scaled the pre-
upgrade magnetograms by a factor of 1.59 to reconcile them to the post-upgrade
magnetograms. However, Pietarila et al. (2013) demonstrated that even with
this step, systematic differences remain between the pre-upgrade and post-
upgrade magnetograms. In another change, the VSM was operated with the
interior filled with helium till January 2014, after which nitrogen was
employed instead. This resulted in a weak but palpable degradation in spatial
resolution (Harvey, 2014). Here, we will refer to VSM before the CCD camera
upgrade as VSM(1), from between this and the switch to nitrogen as VSM(2), and
from after the switch as VSM(3).

###  2.2 Data selection

Considering MDI and VSM observations from before and after the exceptional
instrument events we had just discussed (Sect. 2.1) separately, we make the
following instrument comparisons: MDI(1)-SPM, MDI(2)-SPM, MDI(2)-VSM(1),
MDI(2)-VSM(2), HMI-VSM(2), HMI-VSM(3), and HMI-MDI(2) (Table 2). For each
instrument comparison, for each day on which both instruments recorded
observations, we select the magnetogram from each instrument taken closest in
time to one another. The result is a set of daily near-simultaneous
magnetogram pairs.

Table 2: Overview of the instrument comparisons.

| Number of daily |  |  | FWHMbb$b$The full width at half maximum (FWHM) of the Gaussian kernel applied to the MDI magnetograms in the comparison to SPM and VSM magnetograms, and to the HMI magnetograms in the comparison to VSM and MDI magnetograms (see Sect. 2.3). | Magnetogram
---|---|---|---|---|---
Comparison | magnetogram pairs | Period [year.month.day] | Coverageaa$a$The proportion of the period covered by the daily magnetogram pairs, i.e. the ratio of the number of pairs and the number of days between the first and last pairs. | [arcsec] | noise thresholdcc$c$The magnetogram signal threshold applied to the MDI(1) (first column) and SPM data (second column) in the MDI(1)-SPM comparison, and so on (see Sect. 2.4). [G]
MDI(1)-SPM | 87 | 1996.04.23 - 1998.06.08 | 0.112 | 4.77 | MDI(1): 16.6 | SPM: 13.4
MDI(2)-SPM | 182 | 1999.02.25 - 2003.09.05 | 0.110 | 4.44 | MDI(2): 23.1 | SPM: 15.8
MDI(2)-VSM(1) | 385 | 2003.09.01 - 2009.11.20 | 0.169 | 5.44 | MDI(2): 18.8 | VSM(1): 19.3
MDI(2)-VSM(2) | 104 | 2009.12.18 - 2011.04.11 | 0.217 | 6.34 | MDI(2): 14.9 | VSM(2): 13.7
HMI-VSM(2) | 406 | 2010.05.01 - 2014.01.28 | 0.297 | 5.41 | HMI: 9.09 | VSM(2): 11.6
HMI-VSM(3) | 490 | 2014.02.04 - 2017.10.20 | 0.362 | 6.04 | HMI: 10.6 | VSM(3): 14.0
HMI-MDI(2) | 287 | 2010.04.30 - 2011.04.11 | 0.827 | 3.32 | HMI: 17.6 | MDI(2): 27.3

222

HMI magnetograms exhibit 12-h and 24-h periodicities at high magnetogram
signal levels (Liu et al., 2012). This is an instrumental artefact due to the
combination of strong Zeeman splitting and the diurnal variation in the line-
of-sight (LOS) velocity of the SDO spacecraft. For the HMI-VSM comparisons,
for each day, we select the VSM magnetogram taken closest in time to when the
LOS velocity of SDO relative to the Sun is at its minimum and the HMI
magnetogram taken closest in time to that, similarly for the HMI-MDI
comparison.

We exclude from further consideration the daily magnetogram pairs recorded
more than three hours apart of one another or which suffer severe instrumental
artefacts such as missing values. While MDI and HMI are spaceborne, SPM and
VSM are ground-based and therefore subjected to atmospheric seeing. For the
comparisons involving SPM and VSM, we also exclude the daily magnetogram pairs
where the SPM and VSM magnetograms suffer severe seeing degradation
(identified by visual inspection). In Table 2, we list the number of daily
magnetogram pairs and the period they cover in each of the seven instrument
comparisons.

###  2.3 Data reduction

Our objective is to estimate the uncertainty in measured magnetism due to
changes in instrument response by comparing near-simultaneous observations
from the various instruments. It is essential to account for factors other
than response changes that might affect this comparison. For this reason, we
took the following steps to account for the differences in spatial resolution
and the disparity between two magnetograms recorded some time apart of one
another due to solar rotation. For each instrument comparison, each
magnetogram pair, we rotated the solar disc in time to the midpoint of the
observation times of the two magnetograms. Then, we resampled the magnetograms
from the instrument with the finer pixel scale to the pixel scale of the other
instrument. Finally, we smoothed the set of magnetograms with the higher
spatial resolution (after the resampling in the previous step) to match that
of the other by convolving the former with a Gaussian kernel. The width of the
Gaussian kernel, listed in Table 2, is set at the value that maximises the
correlation between the two sets of magnetograms at disc centre.

For the MDI-SPM and MDI-VSM comparisons, the SPM and VSM magnetograms are
binned to the pixel scale of MDI and the MDI magnetograms are smoothed to the
spatial resolution of the binned SPM and VSM magnetograms. For the HMI-VSM and
HMI-MDI comparisons, the HMI magnetograms are binned and smoothed to the pixel
scale and spatial resolution of VSM and MDI. The width of the Gaussian kernel
differs between the MDI(1)-SPM and MDI(2)-SPM comparisons, between the
MDI(2)-VSM(1) and MDI(2)-VSM(2) comparisons, and between the HMI-VSM(2) and
HMI-VSM(3) comparisons (Table 2). This reflects the fact that the spatial
resolution of MDI and VSM were altered with the exceptional instrument events
discussed in Sect. 2.1.

###  2.4 Magnetogram noise threshold

To distinguish magnetic features in the magnetogram data from noise, we
counted the points on the solar disc where the absolute magnetogram signal
lies above a certain threshold value as magnetic. Let us refer to the
magnetogram signal threshold, determined for each of the two sets of
magnetograms in each instrument comparison, as the noise threshold. Here and
in the rest of the paper, we ignore the points on the solar disc outside the
$\mu=0.2$ loci, where $\mu$ denotes the cosine of the heliocentric angle. (The
excluded annulus is only about $4\%$ of the solar disc by area.)

It is known for SPM, MDI, VSM, and HMI that the magnetogram noise varies with
position on the solar disc (Wenzler et al., 2004; Liu et al., 2012; Pietarila
et al., 2013). For each instrument comparison, each of the two sets of
magnetograms, we determined the noise level as a function of disc position
following the procedure of Ortiz et al. (2002). Taking the magnetogram from a
low-activity day (as indicated by the international sunspot number, Clette et
al., 2016), the 1$\sigma$ noise level at each point on the solar disc is given
by the standard deviation of the points within a square window centred on that
disc position. The window width is set at one-tenth the image width. The
result is a map of the 1$\sigma$ noise level. This is repeated for ten other
low-activity days. We took the median of the eleven noise level maps at each
disc position. This is to minimise any effect of the magnetic activity present
in individual magnetograms. The final estimate of the 1$\sigma$ noise level is
given by the polynomial surface fit to the resultant median map.
Conservatively, we set the noise threshold of the given set of magnetograms at
three times the maximum over the solar disc of the 1$\sigma$ noise level map.
By setting the noise threshold in this manner, we factor out the variation in
noise level with disc position from the subsequent analysis, in particular,
the investigation into the effect of the variation in instrument response with
disc position on measured magnetism (Sect. 3.2).

Let $n_{\rm mag}$ represent, for a given magnetogram, the proportion of the
solar disc where the absolute magnetogram signal is above the noise threshold.
Of course, $n_{\rm mag}$ is systematically higher in the less noisy set of
magnetograms in each instrument comparison. (The less noisy a magnetograph,
the more magnetic structures present on the solar disc can be distinguished
from noise, the greater the value of $n_{\rm mag}$.) It is important for the
purposes of the current study to account for the differences in noise level
between the various instruments for the same reason we needed to account for
the differences in spatial resolution and observation time (Sect. 2.3). To
this end, we raised the noise threshold of the less noisy set of magnetograms
in each instrument comparison such that the mean $n_{\rm mag}$ matches that in
the other set of magnetograms. The final noise threshold is listed in Table 2.
What we have done is to isolate the magnetic activity observable by both
instruments in a given comparison by excluding the weak magnetic signals that
only the less noisy instrument can distinguish from noise.

By applying the same noise threshold to all the magnetograms from a given
instrument in a given comparison, we have implicitly assumed that magnetogram
noise is invariant with time. This is a necessary assumption as it is not
straightforward to map the noise level separately for each individual
magnetogram reliably, at least not by the approach taken here, due to the
confounding influence of the magnetic activity present. The noise level of the
various instruments might have changed over their operational life. This is at
least the case for MDI: Ball et al. (2011) reported that the noise level of
this instrument appears to be higher towards the end of its operational life
than at the beginning. To take this into account, for the MDI(2)-SPM
comparison, we determined the noise threshold of the MDI magnetograms using
observations from close to the end of the period of comparison. We will
discuss the implications of assuming magnetogram noise is invariant with time
in Sect. 3.1.

##  3 Analysis

The aim of this investigation is to address the issue that while various
studies have found evidence that the response of solar magnetographs changes
with time, magnetogram signal level, and disc position, the uncertainty in
measured magnetism due to these instrument fluctuations is not well studied
(Sect. 1). There are two main parts to the analysis. In the first part of the
analysis (Sect. 3.1), we compare the daily solar disc coverage by magnetic
features, $n_{\rm mag}$ in the two sets of magnetograms in each instrument
comparison. From this, we will infer the uncertainty in the longitudinal
magnetogram signal, $B_{\rm los}$ and the disc-integrated photospheric
magnetic flux, $\phi_{\rm tot}$ associated with the time variation in
instrument response. In the second part of the analysis (Sect. 3.2), we will
derive the conversion factor between the two instruments in each comparison as
a function of $B_{\rm los}$ and distance from disc centre. By looking at the
effects of cross-calibrating the two sets of magnetograms with these
conversion factors, we will estimate the uncertainty in $B_{\rm los}$ and
$\phi_{\rm tot}$ from the variation in instrument response with $B_{\rm los}$
and distance from disc centre.

By looking at the variation in instrument response with time and with
magnetogram signal level and disc position separately, what we are doing is to
probe the time variation in instrument response under the assumption that it
is invariant with magnetogram signal level and disc position, and vice versa.
Of course, in reality, instrument response varies with the various factors
simultaneously. However, examining the variation in instrument response with
time, magnetogram signal level and disc position simultaneously is made less
than straightforward by the fact that photospheric magnetic structures are not
evenly distributed in time, field strength or disc position. Active regions
and therefore strong magnetic fields are concentrated in periods of high
activity and at mid-latitudes (e.g. Parnell et al., 2009). For this reason, we
refrain from such an analysis here.

In their analysis of instrument stability, Arge et al. (2002) compared the
total photospheric magnetic flux and open flux indicated by the various
instruments while Riley et al. (2014) examined the global conversion factor as
a function of time (Sect. 1). By the term global we allude to the fact that
Riley et al. (2014) had calculated, for each Carrington rotation, the overall
conversion factor between the synoptic magnetograms from two given
instruments, neglecting the variation in this quantity with magnetogram signal
level and latitude. Due to the greater occurrence of stronger magnetic fields
on the solar surface during periods of enhanced activity and the variation in
instrument response with magnetogram signal level, the total photospheric
magnetic flux, open flux, and global conversion factor can be biased by
activity. This is the reason why we depart from these earlier studies and
instead examine $n_{\rm mag}$, which is unlikely to be biased by activity in
the same way.

###  3.1 Variation with time

In this first part of the analysis, for each instrument comparison, we will
compare $n_{\rm mag}$ in the two sets of magnetograms and determine from it
the uncertainty in $B_{\rm los}$ and $\phi_{\rm tot}$ associated with the time
variation in instrument response. As typical of studies based on longitudinal
magnetograms, including Arge et al. (2002) and Riley et al. (2014), we
approximate the magnetic flux density within each resolution element by
$B_{\rm los}/\mu$. For a particular magnetogram, $\phi_{\rm tot}$ is then
given by the integral of $\left|B_{\rm los}/\mu\right|$ over the solar disc,
excluding the image pixels where $B_{\rm los}$ is below the noise threshold.

Having accounted for the differences in spatial resolution (Sect. 2.3) and
noise level (Sect. 2.4), if two instruments under comparison are stable with
time, then $n_{\rm mag}$ should be, random scatter aside, identical between
the two sets of magnetograms. Conversely, instrument changes can produce
differences in $n_{\rm mag}$. For a particular magnetogram, if the instrument
response at the time of observation is stronger (weaker) than usual such that
magnetogram signals are enhanced (diminished), then $n_{\rm mag}$ could be
biased upwards (downwards). This quantity can also be biased by changes in the
magnetogram noise level and the zero level offset. For this investigation, we
assume magnetogram noise is stable with time and the zero level offset is
negligible, such that we treat any disparity in $n_{\rm mag}$ to be from the
time variation in instrument response alone. We will discuss the implications
of both of these assumptions later in this section.

In Fig. 1, we chart $n_{\rm mag}$ in the two sets of magnetograms in each
instrument comparison (blue and red) and the relative difference, $\Delta
n_{\rm mag}$ (black). Taking the MDI(1)-SPM comparison as an example, on day
$t$,

$\Delta n_{\rm mag}{}_{\rm,MDI(1)-SPM}(t)=2\frac{n_{\rm
mag}{}_{\rm,MDI(1)}(t)-n_{\rm mag}{}_{\rm,SPM}(t)}{n_{\rm
mag}{}_{\rm,MDI(1)}(t)+n_{\rm mag}{}_{\rm,SPM}(t)}.$ (1)

Then, we took the 181-day running mean of $\Delta n_{\rm mag}$, denoted by
$\left<\Delta n_{\rm mag}\right>_{\rm 181d}$, to filter out the random scatter
from effects such as telescope jitter and short-term temperature fluctuations,
drawn in Fig. 2 (solid lines). For a given instrument comparison, the
$\left<\Delta n_{\rm mag}\right>_{\rm 181d}$ on a particular day is given by
the mean $\Delta n_{\rm mag}$ of the daily magnetogram pairs in the 181-day
window centred on that day and this is calculated for each day in the period
of comparison. For the MDI-SPM comparisons, the daily magnetogram pairs cover
only about 10$\%$ of the period of comparison (Table 2). We set the width of
the window at 181 days so that there is sufficient statistics, i.e. a
reasonable number of daily magnetogram pairs in the window. For the other
instrument comparisons, even though the magnetogram data cover more of the
period of comparison, we kept the window width at 181 days. This is so that we
are filtering out the shorter-term variability and isolating the longer-term
variability in $\Delta n_{\rm mag}$ consistently between the various
instrument comparisons. As depicted in Fig. 2, the 181-day running mean of
$\Delta n_{\rm mag}$ reveals this quantity to drift and offset from null.
Following the argument in the previous paragraph, we attribute this to the
time variation in the relative instrument response. As a test, we repeated
this analysis with a window width of 91-day instead. Expectedly, the 91-day
running mean retains more of the shorter-term variability in $\Delta n_{\rm
mag}$ than the 181-day running mean, but the overall trend is largely similar.
This analysis is not particularly sensitive to the window width.

Figure 1: As a function of time, how the various magnetographs compare to one
another. a) The top plot shows, for the MDI(1)-SPM comparison, the proportion
of the solar disc above the magnetogram noise threshold, $n_{\rm mag}$ in the
two sets of magnetograms (blue and red). The bottom plot shows the
proportional difference in $n_{\rm mag}$ between the MDI(1) and SPM
magnetograms, $\Delta n_{\rm mag}{}_{\rm,MDI(1)-SPM}$ (black, Equation 1) and
the proportional change in $B_{\rm los}$ and $\phi_{\rm tot}$ in the MDI(1)
magnetograms from the time variation in instrument response, $\Delta B_{\rm
los}{}_{\rm,MDI(1)-SPM}$ (green, Equation 2) and $\Delta\phi_{\rm
tot}{}_{\rm,MDI(1)-SPM}$ (purple, Equation 3). b) to g) The corresponding
plots for the other instrument comparisons. Due to similarity, the $\Delta
n_{\rm mag}{}$ and $\Delta B_{\rm los}{}$ profiles are largely hidden behind
the $\Delta\phi_{\rm tot}{}$ profiles. See Sect. 3.1. Figure 2: The longer-
term trend in how the various magnetographs compare to one another. For each
instrument comparison, the 181-day running mean of $\Delta n_{\rm mag}$ (solid
lines), $\Delta B_{\rm los}$ (dashed lines), and $\Delta\phi_{\rm tot}$
(dotted lines). The arrows mark the relocation of the SOLIS instrument suite
in 2004 and the replacement of the VSM modulator in 2006.

Going back to the example of the MDI(1)-SPM comparison, let us assume SPM to
be invariant. That is to say, the differences in $n_{\rm mag}$ between the two
sets of magnetograms is from instabilities in the MDI(1) instrument alone. (In
the absence of ideal magnetograms, we have to take one of the two instruments
in each comparison as the reference.) Let us also suppose that at a particular
time, any departure in MDI(1) from the average state of this instrument over
the period of comparison rescales the magnetogram signal recorded at that time
by a factor, denoted $k_{\rm MDI(1)-SPM}$. For a particular daily magnetogram
pair, $k_{\rm MDI(1)-SPM}$ is given by the reciprocal of the factor we have to
multiply the MDI(1) magnetogram by such that the difference in $n_{\rm mag}$
goes to zero. The quantity

$\Delta B_{\rm los}{}_{\rm,MDI(1)-SPM}(t)=k_{\rm MDI(1)-SPM}(t)-1$ (2)

represents the proportional change in the MDI(1) magnetogram signal from the
time variation in instrument response (green, Fig. 1a). The corresponding
proportional change in the disc-integrated photospheric magnetic flux is given
by

$\Delta\phi_{\rm tot}{}_{\rm,MDI(1)-SPM}(t)=\frac{\phi_{\rm
tot}{}_{\rm,MDI(1)-SPM}(t)}{\phi^{*}_{\rm tot,MDI(1)-SPM}(t)}-1,$ (3)

where $\phi_{\rm tot}{}_{\rm,MDI(1)-SPM}$ and $\phi^{*}_{\rm tot,MDI(1)-SPM}$
denote the disc-integrated photospheric magnetic flux in the MDI(1)
magnetogram from day $t$ as it is and after it is rescaled by a factor of
$1/k_{\rm MDI(1)-SPM}$ (purple, Fig. 1a). This analysis is not sensitive to
which instrument we take as the reference in that $\Delta B_{\rm
los}{}_{\rm,MDI(1)-SPM}\approx-\Delta B_{\rm los}{}_{\rm,SPM-MDI(1)}$ and
$\Delta\phi_{\rm tot}{}_{\rm,MDI(1)-SPM}\approx-\Delta\phi_{\rm
tot}{}_{\rm,SPM-MDI(1)}$. The results from the other instrument comparisons
are depicted in Figs. 1b to 1g. As with $\Delta n_{\rm mag}$, we applied a
181-day boxcar filter to $\Delta B_{\rm los}$ and $\Delta\phi_{\rm tot}$ to
filter out the random scatter. The result, $\left<\Delta B_{\rm
los}\right>_{\rm 181d}$ (dashed lines, Fig. 2) and $\left<\Delta\phi_{\rm
tot}\right>_{\rm 181d}$ (dotted lines), trace the drifts and offsets in
$B_{\rm los}$ and $\phi_{\rm tot}$ due to the time variation in instrument
response.

Looking at Fig. 2, with the exception of the MDI(1)-SPM and MDI(2)-VSM(1)
comparisons, the various instruments appear relatively stable with respect to
one another, with $\left<\Delta B_{\rm los}\right>_{\rm 181d}$ and
$\left<\Delta\phi_{\rm tot}\right>_{\rm 181d}$ undulating about null and well
below $5\%$ almost everywhere. In the MDI(1)-SPM comparison (cyan),
$\left<\Delta B_{\rm los}\right>_{\rm 181d}$ and $\left<\Delta\phi_{\rm
tot}\right>_{\rm 181d}$ drifted monotonically, by about $10\%$, over the
period of comparison. The MDI(2)-SPM comparison (pink) does however indicate
that the two instruments are relatively stable with respect to one another
after the SoHO vacation and MDI refocus. In the MDI(2)-VSM(1) comparison
(blue), $\left<\Delta B_{\rm los}\right>_{\rm 181d}$ and
$\left<\Delta\phi_{\rm tot}\right>_{\rm 181d}$ dropped sharply by about $10\%$
in 2004 and rose gradually by a similar margin over the next two years before
they started to exhibit an annual modulation of about $5\%$ to $10\%$. We
attribute the drop in 2004 and the drift over the next two years to the
relocation of the SOLIS instrument suite in 2004 and the VSM modulator
degrading gradually between 2003 and 2006 before being replaced (Pietarila et
al., 2013; Riley et al., 2014), marked in Fig. 2 by the arrows. The strong
annual modulation could allude to unaccounted annual temperature fluctuations.

The root-mean-square (RMS) value of $\left<\Delta B_{\rm los}\right>_{\rm
181d}$ and $\left<\Delta\phi_{\rm tot}\right>_{\rm 181d}$, denoted
$\sigma_{B_{\rm los}{\rm,time}}$ and $\sigma_{\phi_{\rm tot}{\rm,time}}$, is
an estimate of the proportional uncertainty in $B_{\rm los}$ and $\phi_{\rm
tot}$ from the time variation in instrument response, tabulated in Table 3.
Excluding the MDI(1)-SPM and MDI(2)-VSM(1) comparisons, the uncertainty in the
observed LOS magnetic flux density and disc-integrated photospheric magnetic
flux does not exceed $2\%$: $\sigma_{B_{\rm los}{\rm,time}}$ ranged from
$0.6\%$ to $1.6\%$ and $\sigma_{\phi_{\rm tot}{\rm,time}}$ from $0.8\%$ to
$1.9\%$. Of course, the exact value of $\sigma_{B_{\rm los}{\rm,time}}$ and
$\sigma_{\phi_{\rm tot}{\rm,time}}$, calculated from the 181-day running mean
of $\Delta B_{\rm los}$ and $\Delta\phi_{\rm tot}$, depends on the width of
the boxcar filter. The narrower (wider) the boxcar filter, the less (more)
shorter-term variability is filtered out and the higher (lower) the value of
$\sigma_{B_{\rm los}{\rm,time}}$ and $\sigma_{\phi_{\rm tot}{\rm,time}}$. For
example, if we had taken the 91-day running mean instead, $\sigma_{B_{\rm
los}{\rm,time}}$ and $\sigma_{\phi_{\rm tot}{\rm,time}}$ would be a fraction
or few tens of percent higher. Nonetheless, $\sigma_{B_{\rm los}{\rm,time}}$
and $\sigma_{\phi_{\rm tot}{\rm,time}}$ remain useful as indications of the
scale of the uncertainties in $B_{\rm los}$ and $\phi_{\rm tot}$ due to to
instrument instabilities and how the various instrument comparisons match up
to one another in this regard.

Table 3: The uncertainty in measured solar magnetism due to the time variation
in instrument response.

| $\sigma_{B_{\rm los}{\rm,time}}$ | $\sigma_{\phi_{\rm tot}{\rm,time}}$
---|---|---
Comparison | $\left[\%\right]$ | $\left[\%\right]$
MDI(1)-SPM | 3.56 | 6.15
MDI(2)-SPM | 1.57 | 1.94
MDI(2)-VSM(1) | 2.47 | 4.17
MDI(2)-VSM(2) | 1.18 | 1.73
HMI-VSM(2) | 0.561 | 0.813
HMI-VSM(3) | 0.807 | 1.10
HMI-MDI(2) | 0.733 | 1.15
333For each instrument-instrument comparison, the proportional uncertainty in
the LOS magnetic flux density, $\sigma_{B_{\rm los}{\rm,time}}$ and disc-
integrated photospheric magnetic flux, $\sigma_{\phi_{\rm tot}{\rm,time}}$ due
to the time variation in instrument response. The values are scaled by a
factor of 100 to express them as percentages. See Sect. 3.1 for details.

The spatial resolution of MDI and VSM had changed with the exceptional
instrument events discussed in Sect. 2.1, which prompted us to distinguish
between MDI and VSM observations from before and after each event in the
analysis. To elucidate the effect the SoHO vacation and MDI refocus might have
had on the response of this instrument, we combined the MDI(1)-SPM and
MDI(2)-SPM data sets and applied to it the procedure of the MDI(1)-SPM
comparison. We depict the proportional change in the MDI magnetogram signal
due to the time variation in instrument response, $\Delta B_{\rm
los}{}_{\rm,MDI-SPM}$ this yields in Fig. 3a (cyan). We also show the result
of applying the procedure of the MDI(2)-SPM comparison to the combined data
set (pink). If the relative instrument response did not change with the SoHO
vacation and MDI refocus, then $\Delta B_{\rm los}{}_{\rm,MDI-SPM}$ should be
at similar levels before and after this event (dashed line). The SoHO vacation
and MDI refocus is accompanied by a $15\pm 2\%$ offset in $\Delta B_{\rm
los}{}_{\rm,MDI-SPM}$. The MDI magnetograms from before and after this event
can be approximately reconciled to one another by scaling the former by a
factor of 1.15 or the latter by the reciprocal, 0.87. The implicit assumption
here is that SPM is stable over the period of the event, such that the offset
in $\Delta B_{\rm los}{}_{\rm,MDI-SPM}$ is entirely due to changes in the
response of MDI.

Figure 3: The proportional change in magnetogram signal due to the time
variation in instrument response. a) Cyan: The proportional change in the MDI
magnetogram signal, relative to SPM, from the time variation in instrument
response, $\Delta B_{\rm los}{}_{\rm,MDI-SPM}$ derived combining the
MDI(1)-SPM and MDI(2)-SPM data sets and applying the procedure of the
MDI(1)-SPM comparison. Pink: The result from applying the procedure of the
MDI(2)-SPM comparison to the combined data set. b) The estimates of $\Delta
B_{\rm los}{}_{\rm,MDI(2)-VSM}$ from replicating the procedure of the
MDI(2)-VSM(1) (blue) and MDI(2)-VSM(2) comparisons (orange) on the combination
of the data sets from these two comparisons. c) The estimates of $\Delta
B_{\rm los}{}_{\rm,HMI-VSM}$ from replicating the procedure of the HMI-VSM(2)
(green) and HMI-VSM(3) comparisons (red) on the combination of the data sets
from these two comparisons. The colour-coding follows Fig. 2. The dashed lines
mark a) the SoHO vacation and MDI refocus and b) the VSM CCD camera upgrade
and c) filling gas switch. See Sects. 2.1 and 3.1.

In the same manner, we combined the MDI(2)-VSM(1) and MDI(2)-VSM(2) data sets
to examine the effect of the VSM CCD camera upgrade (Fig. 3b) and the HMI-
VSM(2) and HMI-VSM(3) data sets to examine the effect of the VSM filling gas
switch (Fig. 3c). The CCD camera upgrade is accompanied by a $18\pm 2\%$
offset in $\Delta B_{\rm los}{}_{\rm,MDI-VSM}$ (Fig. 3b). Assuming the
response of MDI did not change at the same time, pre-upgrade and post-upgrade
VSM magnetograms can be reconciled to one another by scaling the latter by a
factor of 1.18 or the former by the reciprocal, 0.85. The change in $\Delta
B_{\rm los}{}_{\rm,HMI-VSM}$ at the point of the filling gas switch is
somewhat obscured by the fluctuations in this quantity at other times (Fig.
3c). The degradation in spatial resolution caused by the filling gas switch
does not appear to have a greater effect on the response of VSM than the
instabilities seen in its regular operation. The results obtained for
$\Delta\phi_{\rm tot}$ are similar to what we have shown here for $\Delta
B_{\rm los}$, omitted to avoid repetition.

Next, let us examine the sum effect of the gradual drift in the response of
MDI relative to SPM prior to the SoHO vacation and MDI refocus (i.e. what we
have noted of the MDI(1)-SPM comparison) and of this event itself. Looking at
Fig. 3a, from when MDI started operation to the refocus of this instrument
(dashed line), $\Delta B_{\rm los}{}_{\rm,MDI-SPM}$ rose by $0.24\pm 0.02$
(the sum of the gradual upward drift and the $0.15\pm 0.02\%$ offset after the
SoHO vacation and the MDI refocus). In other words, the MDI magnetogram
signal, relative to the SPM magnetogram signal, increased by $24\pm 2\%$. Of
course, this variation in relative response is likely from changes to both
instruments and not either one alone. Nonetheless, the possibility that the
response of MDI might have changed by any amount close to the rather stark
figure of $24\pm 2\%$ is consequential. The MDI instrument operated from 1996
to 2011, a period which encompasses solar cycle 23 in its entirety. It remains
the only solar telescope to have recorded seeing-free full-disc magnetograms
over a complete solar cycle from minimum to minimum. For this reason, it is a
particularly useful data set to examine how various aspects of the
photospheric magnetic field changes over the solar cycle, such as ephemeral
regions (Hagenaar et al., 2003), the magnetic network (Jin & Wang, 2014) and
the internetwork magnetic field (Meunier, 2018). The observation here that the
response of MDI could have increased significantly over the period of 1996 to
1999, incidentally the ascending phase of solar cycle 23, warrants caution
when interpreting the results from such studies and should be taken into
account in future investigations. For example, Meunier (2018) found the
strength of the internetwork magnetic field, as apparent in MDI observations,
to vary in phase with the solar cycle. However, the rise over the ascending
phase of solar cycle 23 (1996 to 1999) is similar to the change in the
response of MDI relative to SPM over the same period. Consequently, it cannot
be ruled out that this part of their result might, at least in part, be an
instrument artefact. We discuss this in detail in Section 4.2.1.

In the analysis presented in this section, we had assumed that magnetogram
noise is stable with time and the zero level offset is negligible. Recall from
Sect. 2.4, the assumption that magnetogram noise is constant is almost
unavoidable because it is not straightforward to trace its time variation. The
apparent effect of the time variation in instrument response on $B_{\rm los}$
and $\phi_{\rm tot}$ revealed by the above analysis might contain
contributions by noise fluctuations. The influence of instrument instability
could therefore be weaker than estimated here. The residual zero level offset
of the various instruments examined in this study, i.e. what is not already
corrected for by the respective instrument teams, is relatively small (well
below 1 G, Getachew et al., 2019). Also, since the zero level offset would
have the equal and opposite effect on positive and negative magnetic signals,
we expect the effect of the bias in positive and in negative magnetic signals
on $\Delta n_{\rm mag}$, $\Delta B_{\rm los}$, and $\Delta\phi_{\rm tot}$ to
at least partially cancel out. Taking these factors into consideration, any
effect the zero level offset might have had on the above analysis is likely
minute.

It is less than straightforward to compare the results we have obtained here
to what Riley et al. (2014), who had also examine the stability of SPM, MDI,
VSM, and HMI by comparing them to one another, had reported. While we had
inferred the time variation in instrument response from the surface coverage
by magnetic features in full-disc magnetograms, Riley et al. (2014) looked at
the global conversion factor based on magnetic synoptic maps. As noted in the
beginning of Sect. 3, the global conversion factor can be biased by activity.
Since the process of constructing synoptic magnetograms from full-disc
magnetograms evidently involves modifications to the latter, and the various
instrument teams can differ in how this is done, the synoptic magnetograms
from the various instruments might not compare to one another in the same way
as their full-disc magnetograms. A direct comparison is also made difficult by
differences in how the two studies organised the magnetogram data. We split
the MDI and VSM data sets, distinguishing between observations from before and
after the exceptional instrument events detailed in Sect. 3, while Riley et
al. (2014) combined the SPM and VSM data sets into one, which they termed
‘SOLIS’. While the Riley et al. (2014) analysis did indicate fluctuations in
how MDI and ‘SOLIS’ compare with one another, see Fig. 9 in their paper, given
the factors we have just discussed, a direct comparison with what we have
obtained in this study is difficult.

###  3.2 Variation with magnetogram signal level and disc position

In this second part of the study, we examine the effect of the variation in
instrument response with magnetogram signal level and disc position on
measured magnetism. First, we determine the conversion factor between the
various instruments as a function of the absolute magnetogram signal level,
$\left|B_{\rm los}\right|$ and the cosine of the heliocentric angle, $\mu$ and
cross-calibrate the observations from the different instruments with the
result. We neglect the non-radial component of the variation with disc
position, a step we will justify later in this discussion. Then, we repeat
this analysis with the global conversion factor, the overall ratio of the
magnetogram signal registered by two given instruments. Finally, by comparing
the results from the two analyses, we estimate the uncertainty in $B_{\rm
los}$ and $\phi_{\rm tot}$ due to the variation in instrument response with
magnetogram signal level and distance from disc centre. It is worth
emphasising that while earlier investigations have examined the conversion
factor between the various instruments (Sect. 1), this is the first study of
its kind to probe the associated uncertainty in measured magnetism in detail.
As in the first part of the study and for similar reasons (Sect. 3.1), we
assume the zero level offset is negligible.

For each instrument comparison, we determine the conversion factor as a
function of $\left|B_{\rm los}\right|$ and $\mu$ adopting the histogram
equalisation (HE) method (Jones & Ceja, 2001). The idea behind HE is, rather
than comparing the magnetogram signal in simultaneous observations from the
two instruments in question directly, to instead bin the points in the
observations from each instrument by the magnetogram signal and compare the
bin-average magnetogram signal. The procedure here broadly follows Yeo et al.
(2014). Let us take the MDI(1)-SPM comparison as an example. We divide the
solar disc by $\mu$ into 10 intervals, defined such that the number of
magnetic image pixels, i.e. image pixels above the noise threshold (Table 2),
in each interval is roughly even. The interval at disc centre is $0.97<\mu\leq
1.00$. Going from disc centre to limb, the interval width increases steadily
from 0.03 to 0.35, such that the interval closest to the limb is $0.20<\mu\leq
0.55$. We collate the magnetic image pixels present in the two sets of
magnetograms. Suppose a particular SPM magnetogram and the corresponding
MDI(1) magnetogram have, in a given $\mu$-interval, $M$ and $N$ magnetic image
pixels, respectively. If $M<N$, we isolate the $M$ magnetic pixels in the SPM
magnetogram and just the strongest $M$ of the $N$ magnetic pixels in the
MDI(1) magnetogram, and vice versa if $M>N$. This is to factor out the
imbalance between $M$ and $N$ due to factors such as the time variation in
instrument response and the difference in observation time. For each
instrument and interval of $\mu$, we bin the magnetic image pixels collated
from all the magnetograms by $\left|B_{\rm los}\right|$ into 2000 bins of
equal population and calculate the average $\left|B_{\rm los}\right|$ within
each bin. This yields, for each interval of $\mu$, a bin-averaged
$\left|B_{\rm los,MDI(1)}\right|$ series and a bin-averaged $\left|B_{\rm
los,SPM}\right|$ series that together describe the relationship between MDI(1)
and SPM magnetogram signals at this distance from disc centre. Let us refer to
this as the HE table.

In Fig. 4a, we chart the MDI(1)-to-SPM conversion factor, $\left|B_{\rm
los,SPM}\right|/\left|B_{\rm los,MDI(1)}\right|$ in the $0.97<\mu\leq 1.00$
(black solid line), $0.55<\mu\leq 0.64$ (black dashed line), and $0.20<\mu\leq
0.55$ intervals (black dotted line) as a function of $\left|B_{\rm
los,MDI(1)}\right|$. Going from high to low $\left|B_{\rm los,MDI(1)}\right|$,
the MDI(1)-to-SPM conversion factor rises gradually up to $\left|B_{\rm
los,MDI(1)}\right|$ of about 50 G before dropping rapidly at lower
$\left|B_{\rm los,MDI(1)}\right|$ instead. In Figs. 4b to 4g, we depict the
results from the other instrument comparisons. As with the MDI(1)-SPM
comparison, in each instance, going towards the origin, there is an abrupt
change in the trend in the conversion factor with $\left|B_{\rm los}\right|$.
This was similarly noted by Yeo et al. (2014), who argued that at low
magnetogram signal levels, the conversion factor might be biased by noise such
that it no longer indicates the true relationship between the magnetogram
signal scale of the two instruments in question.

Figure 4: The conversion factor between the various magnetographs. a) Black:
The MDI(1)-to-SPM conversion factor as a function of the MDI(1) magnetogram
signal, $\left|B_{\rm los,MDI(1)}\right|$ at $0.97<\mu\leq 1.00$ (solid
curves), $0.55<\mu\leq 0.64$ (dashed curves), and $0.20<\mu\leq 0.55$ (dotted
curves). Red: The corresponding bivariate polynomial in $\left|B_{\rm
los,MDI(1)}\right|$ and $\mu$ fit (Equation 4). The points below three times
the magnetogram noise threshold (blue line) were excluded from the fit. Green:
The global conversion factor (Equation 5). Grey: The conversion factor-
versus-$\left|B_{\rm los,MDI(1)}\right|$ profile in each of the four quadrants
of the solar disc. b) to g) The corresponding plots for the other instrument
comparisons. See Sect. 3.2 for details.

Taking the HE table from the MDI(1)-SPM comparison, we fit a function of the
form

$\left|B_{\rm
los,SPM}\right|=\sum^{3}_{i=1}\sum^{3}_{j=0}c_{ij{\rm,MDI(1)-SPM}}\left|B_{\rm
los,MDI(1)}\right|^{i}\mu^{j},$ (4)

where $c_{ij{\rm,MDI(1)-SPM}}$ are fit parameters (red curves, Fig. 4a). As
the HE table might be biased by magnetogram noise at low $\left|B_{\rm
los}\right|$, noted in the previous paragraph, we exclude the points in the HE
table where $\left|B_{\rm los,MDI(1)}\right|$ is below three times the MDI(1)
noise threshold (blue line) from the regression. What we have done here is to
capture the information in the HE table, which is essentially a lookup table
of how MDI(1) and SPM magnetogram signals relate to one another at each of the
10 intervals of $\mu$, in a relationship that describes the SPM magnetogram
signal as a function of the MDI(1) magnetogram signal and distance from disc
centre. Taking this relationship, we rescale the MDI(1) magnetograms to the
SPM magnetogram signal scale. In the same manner, we cross-calibrate the two
sets of magnetograms in each of the other six instrument comparisons. To aid
the following discussion, let us refer to this analysis as case A.

Next, we repeat the above analysis with the global conversion factor. Going
back to the example of the MDI(1)-SPM comparison, we derive the HE table as
before, except taking the entire solar disc as a whole. To the scatter plot of
the bin-average $\left|B_{\rm los,SPM}\right|$ and $\left|B_{\rm
los,MDI(1)}\right|$ values in the HE table, we fit a straight line that is
constrained to pass through the origin. In other words, we assume

$\left|B_{\rm los,SPM}\right|=m_{B_{\rm los}{\rm,MDI(1)-SPM}}\left|B_{\rm
los,MDI(1)}\right|,$ (5)

where the fit parameter $m_{B_{\rm los}{\rm,MDI(1)-SPM}}$ indicates the global
MDI(1)-to-SPM conversion factor (green line, Fig. 4a). For the same reason as
in case A, we exclude the points in the HE table where $\left|B_{\rm
los,MDI(1)}\right|$ is below three times the MDI(1) noise threshold from the
regression. Finally, we rescaled the MDI(1) magnetograms by $m_{B_{\rm
los}{\rm,MDI(1)-SPM}}$. Let us refer to this analysis as case B.

Various studies have examined the conversion factor between the same
instruments we are looking at here (Jones & Ceja, 2001; Wenzler et al., 2004;
Demidov et al., 2008; Liu et al., 2012; Pietarila et al., 2013; Riley et al.,
2014; Yeo et al., 2014). The objective of the current study is not to
investigate the conversion factor, but rather to use this information to
estimate the effect of the variation in instrument response with magnetogram
signal level and distance from disc centre on measured magnetism. In Sect.
4.1, we will compare the conversion factors derived here with what was
reported in the earlier efforts. Otherwise, we refer the reader to the cited
studies for more detailed discussions of the conversion factor between the
various instruments, including the instrumental differences that underlie the
observed values.

Let us examine the effect of the cross-calibration. For each instrument
comparison, taking the HE table derived in case A by binning the points in the
magnetogram observations by $\left|B_{\rm los}\right|$ and by $\mu$ and taking
the bin-average, we calculate the RMS of the relative difference between the
$\left|B_{\rm los}\right|$ registered by the two instruments, $\sigma_{B_{\rm
los}}$. Taking the example of the MDI(1)-SPM comparison, this is given by

$\sigma_{B_{\rm
los}{\rm,MDI(1)-SPM}}=\sqrt{\frac{1}{n}\sum\left(2\frac{\left|B_{\rm
los,MDI(1)}\right|-\left|B_{\rm los,SPM}\right|}{\left|B_{\rm
los,MDI(1)}\right|+\left|B_{\rm los,SPM}\right|}\right)^{2}},$ (6)

where the summation is over the $n=20000$ bin-average values in the HE table.
The quantity $\sigma_{B_{\rm los}}$ is an estimate of the proportional
uncertainty in $B_{\rm los}$ due to the systematic differences between the two
sets of magnetograms under comparison. In a similar manner, we estimate the
proportional uncertainty in $\phi_{\rm tot}$, $\sigma_{\phi_{\rm tot}}$ by
calculating the RMS of the relative difference between $\phi_{\rm tot}$ in the
two sets of magnetograms in each instrument comparison. This is repeated for
the cross-calibrated data from case A and case B. Let us denote the result of
repeating this computation with the cross-calibrated data from case A as
$\sigma_{B_{\rm los}{\rm,A}}$ and $\sigma_{\phi_{\rm tot}{\rm,A}}$ and
likewise that from case B as $\sigma_{B_{\rm los}{\rm,B}}$ and
$\sigma_{\phi_{\rm tot}{\rm,B}}$. The results are tabulated in Table 4. We
note here that this analysis is commutative in that the results from switching
the roles of the two instruments in each comparison, e.g. rescaling SPM
magnetograms to the MDI magnetogram signal scale instead of vice versa, are
similar.

Table 4: The uncertainty in measured solar magnetism due to the systematic
differences between the various magnetographs.

| $\sigma_{B_{\rm los}}$ | $\sigma_{B_{\rm los}{\rm,A}}$ | $\sigma_{B_{\rm los}{\rm,B}}$ | $\sigma_{B_{\rm los}{\rm,mag,ddc}}$
---|---|---|---|---
Comparison | $\left[\%\right]$ | $\left[\%\right]$ | $\left[\%\right]$ | $\left[\%\right]$
MDI(1)-SPM | 40.7 | 2.39 | 8.08 | 7.71
MDI(2)-SPM | 23.5 | 5.28 | 14.4 | 13.4
MDI(2)-VSM(1) | 5.51 | 2.87 | 9.37 | 8.92
MDI(2)-VSM(2) | 39.7 | 2.52 | 13.9 | 13.7
HMI-VSM(2) | 26.2 | 5.19 | 12.6 | 11.5
HMI-VSM(3) | 10.8 | 3.59 | 11.8 | 11.3
HMI-MDI(2) | 19.6 | 4.36 | 14.0 | 13.3
| $\sigma_{\phi_{\rm tot}}$ | $\sigma_{\phi_{\rm tot}{\rm,A}}$ | $\sigma_{\phi_{\rm tot}{\rm,B}}$ | $\sigma_{\phi_{\rm tot}{\rm,mag,ddc}}$
Comparison | $\left[\%\right]$ | $\left[\%\right]$ | $\left[\%\right]$ | $\left[\%\right]$
MDI(1)-SPM | 20.2 | 7.97 | 24.8 | 23.4
MDI(2)-SPM | 39.9 | 5.00 | 12.1 | 11.0
MDI(2)-VSM(1) | 7.91 | 8.13 | 15.2 | 12.9
MDI(2)-VSM(2) | 13.6 | 5.81 | 13.0 | 11.6
HMI-VSM(2) | 14.2 | 5.68 | 9.18 | 7.21
HMI-VSM(3) | 21.5 | 8.76 | 14.3 | 11.3
HMI-MDI(2) | 36.6 | 5.01 | 8.62 | 7.02
444Top: For each instrument comparison, the proportional uncertainty in
$B_{\rm los}$ in the original data, $\sigma_{B_{\rm los}}$ and in the cross-
calibrated data from case A, $\sigma_{B_{\rm los}{\rm,A}}$ and case B,
$\sigma_{B_{\rm los}{\rm,B}}$. The proportional uncertainty in $B_{\rm los}$
due to the variation in instrument response with magnetogram signal level and
distance from disc centre, $\sigma_{B_{\rm los}{\rm,mag,ddc}}$ is also listed.
Bottom: The same for the proportional uncertainty in the disc-integrated
photospheric magnetic flux, $\sigma_{\phi_{\rm tot}}$. The values are scaled
by a factor of 100 to express them as percentages. See Sect. 3.2 for details.

Looking at Table 4, the case B cross-calibration brought $B_{\rm los}$ and
$\phi_{\rm tot}$ in the two sets of magnetograms in each instrument comparison
closer together, i.e. $\sigma_{B_{\rm los}{\rm,B}}<\sigma_{B_{\rm los}}$ and
$\sigma_{\phi_{\rm tot}{\rm,B}}<\sigma_{\phi_{\rm tot}}$, in every instance
except the following. For the MDI(2)-VSM(1) and HMI-VSM(3) comparisons,
$\sigma_{B_{\rm los}{\rm,B}}>\sigma_{B_{\rm los}}$ and for the MDI(1)-SPM and
MDI(2)-VSM(1) comparisons, $\sigma_{\phi_{\rm tot}{\rm,B}}>\sigma_{\phi_{\rm
tot}}$. The case A cross-calibration brought $B_{\rm los}$ and $\phi_{\rm
tot}$ in the two sets of magnetograms in each instrument comparison closer
together except in the MDI(2)-VSM(1) comparison, where $\sigma_{\phi_{\rm
tot}{\rm,A}}>\sigma_{\phi_{\rm tot}}$. Comparing case A and case B, without
exception, $\sigma_{B_{\rm los}{\rm,A}}<\sigma_{B_{\rm los}{\rm,B}}$ and
$\sigma_{\phi_{\rm tot}{\rm,A}}<\sigma_{\phi_{\rm tot}{\rm,B}}$. As was to be
expected, accounting for the rather significant variation in the conversion
factor with $\left|B_{\rm los}\right|$ and $\mu$ (Fig. 4) in the cross-
calibration (case A) reconciles the observations from the various instruments
more closely than if it were ignored (case B). This limitation of the case B
analysis and the observation in Sect. 3.1 that the MDI(1)-SPM and
MDI(2)-VSM(1) comparisons are particularly affected by instrument
instabilities are likely, at least in part, why $\sigma_{B_{\rm los}}$ and
$\sigma_{\phi_{\rm tot}}$ are higher in the cross-calibrated data than in the
original data in the instances we have highlighted above.

The proportional uncertainty in $B_{\rm los}$ in the original data,
$\sigma_{B_{\rm los}}$, is the result of all the factors that contributed to
the systematic differences between the two sets of magnetograms under
comparison. Let $\sigma_{B_{\rm los},1}$, $\sigma_{B_{\rm los},2}$,
$\sigma_{B_{\rm los},3}$, and so on represent the proportional uncertainty in
$B_{\rm los}$ due to individual factors. Let us suppose that the different
contributions to the uncertainty in $B_{\rm los}$ combine in quadrature, such
that

$\sigma_{B_{\rm los}}^{2}=\sigma_{B_{\rm los},1}^{2}+\sigma_{B_{\rm
los},2}^{2}+\sigma_{B_{\rm los},3}^{2}+\ldots$ (7)

Cross-calibration can be thought of as the process of accounting for certain
factors, removing the terms corresponding to them from the right hand side of
Equation 7. The square root of the sum of the remaining terms, which
correspond to the factors that remain unaccounted for, yields $\sigma_{B_{\rm
los}}$ in the cross-calibrated data. In case B, by the cross-calibration with
the global conversion factor, we accounted for the differences in absolute
scale between the various instruments. Following the argument we had just
presented,

$\sigma_{B_{\rm los}}^{2}=\sigma_{B_{\rm los}{\rm,B}}^{2}+\sigma_{B_{\rm
los}{\rm,abs}}^{2},$ (8)

where $\sigma_{B_{\rm los}{\rm,abs}}$ represents the proportional uncertainty
in $B_{\rm los}$ associated with the uncertainty in the absolute scale. In
case A, by determining the conversion factor as a function of $\left|B_{\rm
los}\right|$ and $\mu$, we additionally accounted for the variation in
instrument response with magnetogram signal level and distance from disc
centre, such that

$\sigma_{B_{\rm los}}^{2}=\sigma_{B_{\rm los}{\rm,A}}^{2}+\sigma_{B_{\rm
los}{\rm,abs}}^{2}+\sigma_{B_{\rm los}{\rm,mag,ddc}}^{2},$ (9)

where $\sigma_{B_{\rm los}{\rm,mag,ddc}}$ represents the proportional
uncertainty in $B_{\rm los}$ due to this factor. Combining Equations 8 and 9,

$\sigma_{B_{\rm los}{\rm,mag,ddc}}=\sqrt{\sigma_{B_{\rm
los}{\rm,B}}^{2}-\sigma_{B_{\rm los}{\rm,A}}^{2}}.$ (10)

By the same argument, the proportional uncertainty in $\phi_{\rm tot}$ due to
the variation in instrument response with magnetogram signal level and
distance from disc centre, $\sigma_{\phi_{\rm tot}{\rm,mag,ddc}}$ is given by

$\sigma_{\phi_{\rm tot}{\rm,mag,ddc}}=\sqrt{\sigma_{\phi_{\rm
tot}{\rm,B}}^{2}-\sigma_{\phi_{\rm tot}{\rm,A}}^{2}}.$ (11)

Taking Equations 10 and 11, we calculate $\sigma_{B_{\rm los}{\rm,mag,ddc}}$
and $\sigma_{\phi_{\rm tot}{\rm,mag,ddc}}$, tabulated in Table 4.

The variation in instrument response with magnetogram signal level and
distance from disc centre produced an overall uncertainty of 8$\%$ to 14$\%$
in $B_{\rm los}$ and of 7$\%$ to 23 $\%$ in $\phi_{\rm tot}$ ($\sigma_{B_{\rm
los}{\rm,mag,ddc}}$ and $\sigma_{\phi_{\rm tot}{\rm,mag,ddc}}$, Table 4). This
rather significant uncertainty is why the cross-calibration with the global
conversion factor, which neglects this variability, only reconciled the
$B_{\rm los}$ registered by the various instruments to within about 5$\%$ to
9$\%$ and $\phi_{\rm tot}$ to 9$\%$ to 25$\%$ ($\sigma_{B_{\rm los}{\rm,B}}$
and $\sigma_{\phi_{\rm tot}{\rm,B}}$). Accounting for the variation in
instrument response with magnetogram signal level and distance from disc
centre enabled us to reconcile $B_{\rm los}$ to within 2$\%$ to 5$\%$ and
$\phi_{\rm tot}$ to within 5$\%$ to 9$\%$ ($\sigma_{B_{\rm los}{\rm,A}}$ and
$\sigma_{\phi_{\rm tot}{\rm,A}}$), demonstrating the importance in taking it
into account when comparing observations from multiple solar magnetographs.
For each instrument comparison, both $\sigma_{B_{\rm los}{\rm,A}}$ and the
uncertainty in $B_{\rm los}$ due to the time variation in instrument response,
$\sigma_{B_{\rm los}{\rm,time}}$ (Table 3) are smaller than $\sigma_{B_{\rm
los}{\rm,mag,ddc}}$, and likewise for $\phi_{\rm tot}$. This suggests that the
uncertainty in measured magnetism and the systematic differences between the
observations from the various instruments is dominated by the variation in
instrument response with magnetogram signal level and distance from disc
centre.

We had confined this analysis to the radial component of the variation in
instrument response with disc position even though the instrument response at
a given distance from disc centre might also vary with polar angle. It is less
than straightforward to determine the conversion factor as a function of
$\left|B_{\rm los}\right|$, $\mu$, and polar angle simultaneously. One
obstacle is the fact that higher magnetogram signals are concentrated at mid-
latitudes. However, as we will demonstrate next, the variation in instrument
response with polar angle is likely small compared to the variation with
magnetogram signal level and distance from disc centre, such that its omission
here is not detrimental to the objectives of the study.

In the case A analysis, we had divided the solar disc by $\mu$ into 10
intervals and derived the conversion factor as a function of $\left|B_{\rm
los}\right|$ within each interval (black curves, Fig. 4). We repeat this
computation, except now we divide the solar disc by the central meridian and
the east-west line crossing the disc centre into four quadrants instead. The
conversion factor-versus-$\left|B_{\rm los}\right|$ profile within each
quadrant are drawn in grey, alongside the earlier results. The offset between
the various profiles is due to the quadrant-to-quadrant differences in
instrument response and distribution of magnetic image pixels with $\mu$. As
is visibly evident, for the various instrument comparisons, this offset is
relatively small as compared to the variation in the conversion factor with
$\left|B_{\rm los}\right|$ and $\mu$. Within the limits of this analysis,
there is little evidence that the variation in instrument response with polar
angle might be significant.

##  4 Discussion

###  4.1 Comparison between the conversion factors from the current study and
earlier studies

In Sect. 3.2, for the purpose of quantifying the uncertainty in measured
magnetism due to the variation in instrument response with magnetogram signal
level and distance from disc centre, we derived, for each instrument
comparison, the conversion factor as a function of $B_{\rm los}$ and $\mu$
(black curves, Fig. 4) and also the global conversion factor (green lines). In
this section, we discuss the earlier studies that had examined the conversion
factor between the same instruments we are looking at here (Jones & Ceja,
2001; Wenzler et al., 2004; Demidov et al., 2008; Liu et al., 2012; Pietarila
et al., 2013; Riley et al., 2014; Yeo et al., 2014).

Jones & Ceja (2001), Wenzler et al. (2004), and Demidov et al. (2008) examined
the conversion factor between SPM and MDI. Jones & Ceja (2001) found a value
of 0.7 for the MDI(2)-to-SPM global conversion factor. Demidov et al. (2008)
found a value of 0.9 for the MDI(2)-to-SPM global conversion factor. They also
determined the MDI(2)-to-SPM conversion factor as a function of $B_{\rm los}$:
going from low to high $B_{\rm los}$, the conversion factor declines from 1.2
to 0.8. (We have divided the values reported by the authors by a factor of
1.46 to account for the fact that they had scaled the SPM magnetograms by this
factor prior to analysis.)

In this study, we retrieved a value of 0.6 for the MDI(2)-to-SPM global
conversion factor (green line, Fig. 4b). As for the conversion factor
determined as a function of $B_{\rm los}$ and $\mu$, going from low to high
$B_{\rm los}$, it decreases from 0.7 to 0.5 around disc centre (black solid
line) and from 0.9 to 0.8 near the limb (black dotted line). The values we
have obtained are a fraction lower than what Jones & Ceja (2001) and Demidov
et al. (2008) had reported, likely due to the following. We had resampled the
SPM magnetograms to the MDI pixel scale and smoothed the MDI magnetograms with
a Gaussian kernel to account for the difference in spatial resolution (Sect.
2.3). Jones & Ceja (2001) and Demidov et al. (2008) had similarly resampled
the SPM magnetograms but kept the MDI magnetograms as they are, therefore only
partially accounted for the difference in spatial resolution. We cannot make a
direct comparison with the Wenzler et al. (2004) study. In their analysis, the
authors had combined MDI observations from before and after the SoHO vacation
and the refocus of this instrument. Also, they had examined the SPM and MDI
data at their original spatial resolution.

Counting image pixels with $B_{\rm los}$ below and above a certain threshold
level as weak and strong fields, Liu et al. (2012) calculated the HMI-to-
MDI(2) conversion factor for strong and weak fields separately and within
various intervals of $\mu$, returning values ranging from 1.1 to 1.5 (see
Tables 1 and 2 in their study). The conversion factor is higher around disc
centre and for weak fields than near the limb and for strong fields. In a
similar analysis, Pietarila et al. (2013) derived MDI(2)-to-VSM(1) conversion
factors of 0.8 and 1.0 for strong and weak fields, respectively, and HMI-to-
VSM(2) conversion factors of 0.7 and 1.0 for the same categories.

From this study, the HMI-to-MDI(2) conversion factor, determined as a function
of $B_{\rm los}$ and $\mu$, ranges from 1.0 to 1.6 (Fig. 4g), which
encompasses the range of values reported by Liu et al. (2012). Importantly,
consistent with this earlier study, the conversion factor decreases from disc
centre to limb and from low to high $B_{\rm los}$. The MDI(2)-to-VSM(1)
conversion factor ranges from 0.8 to 1.1 (Fig. 4c) and the HMI-to-VSM(2)
conversion factor from 0.8 to 1.4 (Fig. 4e), broadly consistent with the
values reported by Pietarila et al. (2013). In both instances, the conversion
factor increases with decreasing $B_{\rm los}$, similar to this earlier study.
Keeping in mind that Liu et al. (2012) and Pietarila et al. (2013) had
partitioned the magnetogram data into broader intervals of $B_{\rm los}$ and
$\mu$ than what we have done here, a close match to the current results is not
expected.

We cannot make a direct comparison with Riley et al. (2014) and Yeo et al.
(2014). Comparing the conversion factors we have derived as part of the
analysis presented in Sect. 3.2 to what Riley et al. (2014) reported is less
than straightforward for the same reasons why we could not compare the
analysis presented in Sect. 3.1 to this earlier study, discussed in that
section. Similar to what we have done here, Yeo et al. (2014) derived the
conversion factor between SPM, MDI, and HMI as a function of $B_{\rm los}$ and
$\mu$. However, while we had resampled and smoothed the magnetogram data from
the various instruments as necessary to account for the differences in spatial
resolution (Sect. 2.3), Yeo et al. (2014) examined them at their original
spatial resolution, which obviously leads to different conversion factors.

###  4.2 Application of the results from the current study to earlier studies

In this study, we had quantified the uncertainty in measured magnetism due to
changes in instrument response, results that are useful to the interpretation
of SPM, MDI, VSM, and HMI magnetograms. As examples, let us discuss the
implications of the current findings on the investigations by Meunier (2018)
and Yeo et al. (2014).

####  4.2.1 Meunier (2018)

Taking MDI magnetograms from the period of 1996 to 2010, Meunier (2018)
calculated the mean $\left|B_{\rm los}\right|/\mu$ over the quietest region
near disc centre, $\left<\left|B_{\rm los}\right|/\mu\right>_{\rm in}$, which
they interpreted as a measure of the internetwork magnetic field. From 1996 to
1999, $\left<\left|B_{\rm los}\right|/\mu\right>_{\rm in}$ increases steadily
from about 6 G to 7.5 G, from 1999 to 2001 it is roughly constant, from 2001
to 2004 it decreases steadily from 7.5 G to 6.5 G and thereafter, it remains
roughly constant again (see Fig. 2D in their paper). The observation that
$\left<\left|B_{\rm los}\right|/\mu\right>_{\rm in}$ is lower during the 1996
and 2008 solar cycle minima and higher during the 2001 maximum suggests that
the internetwork magnetic field varies in phase with the solar cycle, in
contradiction to various studies which concluded otherwise (Lites, 2011;
Buehler et al., 2013; Rempel, 2014; Lites et al., 2014).

The current study revealed that over the period of 1996 to 1999, the response
of MDI relative to SPM had risen such that, as compared to SPM, the MDI
magnetogram signal appear to be biased upwards by $24\pm 2\%$, discussed in
Sect. 3.1. As noted then, this is likely the sum manifest of response changes
in both instruments and not either instrument alone. The SoHO vacation and the
MDI refocus, which occurred during this period, is accompanied by a bias of
$15\pm 2\%$, suggesting that at least this much of the total bias of $24\pm
2\%$ might be due to MDI alone. In other words, the response of MDI changed in
this period such that its magnetogram signal is biased upwards by at least 15%
and possibly by as much as 24%. Comparing this to the 25% increase in
$\left<\left|B_{\rm los}\right|/\mu\right>_{\rm in}$ over the same period
reported by Meunier (2018), we conjecture that what the author noted is at
least in part, if not completely, an instrumental artefact.

####  4.2.2 Yeo et al. (2014)

Yeo et al. (2014) reconstructed the variation in total and spectral solar
irradiance, TSI and SSI, since 1974 using a model based on full-disc
magnetograms. They made use of full-disc magnetograms from the KPVT
512-channel magnetograph (512CM, Livingston et al., 1976), SPM, MDI(2) (i.e.
observations from after the SoHO vacation and the refocus of this instrument),
and HMI. In order for the model output based on the observations from the
various instruments to be mutually consistent, Yeo et al. (2014) rescaled the
MDI(2) magnetograms to the HMI magnetogram signal scale with the conversion
factor determined as a function of $B_{\rm los}$ and $\mu$, similar to the
case A analysis we had presented in Sect. 3.2. The SPM magnetograms were
similarly cross-calibrated to the rescaled MDI(2) magnetograms and the 512CM
magnetograms to the rescaled SPM magnetograms.

The solar irradiance reconstruction presented by Yeo et al. (2014) is based on
SPM magnetograms for the period of 1992 to 2003 and MDI(2) magnetograms for
the period of 1999 to 2010. The reconstruction indicates that between the 1996
and 2008 solar cycle minima, TSI declined by $0.24\ {\rm Wm^{-2}}$.
Considering the fact that the variation in TSI over the solar cycle is around
$1\ {\rm Wm^{-2}}$, this is significant. Making use of the results obtained in
the current study, let us examine the likelihood that the minimum-to-minimum
decline indicated by the reconstruction might be an artefact of systematic
differences between the 1996 SPM magnetograms and the 2008 MDI(2)
magnetograms.

The Yeo et al. (2014) model describes the contribution to solar irradiance
variability by faculae, identified in the magnetograms, and by sunspots,
identified in concurrent continuum images. In the Yeo et al. (2014) model, to
an approximation, a 10% bias in $\left|B_{\rm los}\right|$ is accompanied by a
bias of $0.08\ {\rm Wm^{-2}}$ in the TSI reconstruction. For the minimum-to-
minimum decline of $0.24\ {\rm Wm^{-2}}$ in the TSI reconstruction to have
come from a bias in the 1996 SPM magnetograms relative to the 2008 MDI(2)
magnetograms, this bias has to render 1996 SPM magnetogram signals 30%
stronger.

Let us examine if the cross-calibration between SPM and MDI(2) and the time
variation in instrument response might have produced such a bias. As noted in
the beginning of this section, the Yeo et al. (2014) cross-calibration of SPM
and MDI(2) magnetograms is broadly similar to the case A analysis presented in
Sect. 3.2, where we determined the uncertainty in $B_{\rm los}$ due to the
cross-calibration, $\sigma_{B_{\rm los}{\rm,A}}$, to be about 5% (Table 4).
This being the case, we expect Yeo et al. (2014) to have reconciled the
magnetogram signal registered by these two instruments to within a similar
margin. Looking at Fig. 2, at the instrument comparisons that included either
SPM or MDI(2), the quantity $\Delta B_{\rm los}$ (dashed lines), which traces
the relative bias in $B_{\rm los}$ due to the time variation in instrument
response, is at most times below 5%. For the SPM-MDI(1) comparison, drawn in
cyan, there is a steady drift in $\Delta B_{\rm los}$ of about 10% over the
period of 1996 to 1998. For the sake of argument, let us suppose that MDI(1)
had been stable over this period such that this steady drift is due to SPM
alone. This would mean that relative to the 1998 SPM magnetograms, the 1996
SPM magnetograms are biased such that magnetogram signals are 10% stronger.
The cross-calibration and the time variation in instrument response could have
contributed to the minimum-to-minimum decline in the TSI reconstruction, but
none of these factors appear strong enough (being well below 30%) to fully
account for this drop. We emphasis that this is not a validation of the Yeo et
al. (2014) finding. All this discussion shows is that the minimum-to-minimum
drop is unlikely, at least not predominantly, from a bias in the 1996 SPM
magnetograms relative to the 2008 MDI(2) magnetograms.

##  5 Summary

The utility of the full solar disc magnetograms recorded over the past half-
century as a long-term record of photospheric magnetism depends on an
understanding of how stable these observations are with time and the
systematic differences between the various instruments. While studies
comparing the observations from the various solar telescopes have found
evidence of fluctuations in instrument response with time, magnetogram signal
level, and disc position, the effect on measured magnetism has not been
examined in detail (Sect. 1). The aim of this study is to address this issue.
Comparing full-disc magnetograms from the KPVT/SPM (Jones et al., 1992),
SoHO/MDI (Scherrer et al., 1995), SOLIS/VSM (Keller et al., 2003), and SDO/HMI
(Schou et al., 2012), we quantified the uncertainty in the longitudinal
magnetogram signal, $B_{\rm los}$ and disc-integrated photospheric magnetic
flux, $\phi_{\rm tot}$ due to the variation in instrument response with time,
magnetogram signal and distance from disc centre.

For each instrument-to-instrument comparison, we selected magnetograms the two
instruments recorded close in time to one another and from over the entire
period both of them operated (Sect. 2.2). The data were processed in such a
way as to factor out the differences in observation time, spatial resolution,
and magnetogram noise level (Sect. 2.3 and 2.4). We compared the surface
coverage by magnetic features in the two sets of magnetograms and found the
bias we have to introduce to the magnetograms from one of the two instruments
in order to offset the difference. This allowed us to trace the time variation
in instrument response in terms of its effect on measured $B_{\rm los}$ and
$\phi_{\rm tot}$ (Sect. 3.1). This approach circumvents certain limitations of
earlier attempts to examine such instrument changes (Sect. 3). We derived the
conversion factor, the ratio of the magnetogram signal registered by the two
instruments, as a function of $B_{\rm los}$ and the cosine of the heliocentric
angle, $\mu$. Taking this result, we rescaled the magnetograms from one
instrument to the magnetogram signal scale of the other. We repeated this
analysis with the global conversion factor, in other words, the overall ratio
derived ignoring the $B_{\rm los}$ and $\mu$-dependence. By comparing the
results from the two cross-calibrations, we estimated the uncertainty in
$B_{\rm los}$ and $\phi_{\rm tot}$ due to the variation in instrument response
with magnetogram signal level and distance from disc centre (Sect. 3.2).
Earlier studies have examined the conversion factor between the same
instruments we looked at here, reviewed in Sect. 4.1, and the study by Riley
et al. (2014) even made use of this quantity to trace the time variation in
instrument response. This study extends these earlier works by examining the
effect of instrument response changes on measured magnetism in detail.

Between 1998 and 1999, the SoHO spacecraft suffered a series of extended
outages, often referred to as the SoHO vacation, at the end of which the MDI
instrument was refocused. From 1996, when MDI started operation, right up to
the SoHO vacation, the response of MDI, relative to SPM, drifted steadily
(Figs. 1a and 2). The SoHO vacation and MDI refocus is accompanied by a
further offset in the relative response (Fig. 3a). The result is that as
compared to SPM, the MDI magnetogram signal appear to be biased upwards by
24±2% over the period of 1996 to 1999, with 15±2% coming from the SoHO
vacation and MDI refocus alone. By this estimation, MDI magnetograms from
before and after the SoHO vacation and MDI refocus can be reconciled to one
another by scaling the former by a factor of $1.15\pm 0.02$. After the SoHO
vacation and MDI refocus, MDI and SPM appear to be relatively stable with
respect to one another (Figs. 1b and 2).

The VSM-to-MDI comparison indicted that the response of VSM might have changed
with the relocation of the SOLIS instrument suite in 2004, the modulator
degrading from 2003 to 2006 before its replacement, and the CCD camera upgrade
in 2009 (Figs. 2 and 3b). This is such that as compared to MDI, the VSM
magnetogram signal appears to be biased by about 10% to 20% with each of these
events. From this analysis, we estimated that VSM magnetograms from before the
CCD camera upgrade can be cross-calibrated to the post-upgrade observations by
scaling them by a factor of $1.18\pm 0.02$. When VSM switched from filling its
interior with helium to using nitrogen instead in 2014, the spatial resolution
degraded slightly. The VSM-to-HMI comparison indicated that the change in the
response of VSM relative to HMI this produced appears to be similar in
magnitude to the fluctuations that occurred at other times in the regular
operation of both instruments (Fig. 3c).

Apart from what we have just noted about MDI and VSM, the various instruments
appear to be relatively stable with respect to one another (Fig. 2). The
uncertainty in $B_{\rm los}$ and $\phi_{\rm tot}$ due to the time variation in
instrument response is less than 2% (Table 3). From the cross-calibration
analysis, we estimated the uncertainty in $B_{\rm los}$ due to the variation
in instrument response with magnetogram signal level and distance from disc
centre to be about 8$\%$ to 14$\%$ and in $\phi_{\rm tot}$ to be 7$\%$ to 23
$\%$ (Table 4). (We did not examine in detail the variation in instrument
response with polar angle, i.e. the non-radial component of the variation with
disc position, but we did find evidence that it is likely minute in
comparison, see Fig. 4.) Taking the variation in instrument response with
magnetogram signal and distance from disc centre into account in the cross-
calibration brought the various instruments to with 2% to 5% of one another in
terms of $B_{\rm los}$ and 5% to 9% in terms of $\phi_{\rm tot}$. This
residual and the uncertainty due to the time variation in instrument response
are smaller than the uncertainty due to the variation in instrument response
with magnetogram signal level and distance from disc centre, suggesting that
the latter dominates the uncertainty in measured magnetism and the systematic
differences between the observations from the various instruments.

The results obtained here are useful for the interpretation of SPM, MDI, VSM,
and HMI magnetograms, whether a reassessment of earlier results obtained with
such data or to inform future studies. As examples, we applied our findings to
certain results from Meunier (2018) and Yeo et al. (2014), detailed in Sect.
4.2. Meunier (2018) reported a rise in the strength of the internetwork
magnetic field, as apparent in MDI observations, over the period of 1996 to
1999. We argued it cannot be excluded that this trend is, at least in part, an
artefact of the changes in the response of MDI over this period. The model
reconstruction of total solar irradiance by Yeo et al. (2014) indicates a
decline in this quantity between the 1996 and 2008 solar cycle minima. The
1992 to 2010 segment of the reconstruction is based on cross-calibrated SPM
and MDI magnetograms. From the results of the current study, we argued that
the minimum-to-minimum drop reported by Yeo et al. (2014) is unlikely to be,
at least not predominantly, an artefact of the cross-calibration or the time
variation in the response of either instrument.

###### Acknowledgements.

The authors are grateful to Jack Harvey and Andrew Marble for the useful
discussions. This work made use of magnetogram observations from KPVT/SPM,
SoHO/MDI, SOLIS/VSM, and SDO/HMI. This work was supported by the German
Federal Ministry of Education and Research (Project No. 01LG1909C) and the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (Grant Agreement No. 695075).

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