11institutetext: Institute for Solar Physics (KIS), Schönek Str. 6, 79111
Freiburg, Germany
11email<EMAIL_ADDRESS>22institutetext: Faculty of Mathematics,
University of Belgrade, Studentski Trg 16, 11000 Belgrade, Serbia
33institutetext: High Altitude Observatory (NCAR), 3080 Center Green Dr.,
Boulder, CO, 80301, USA 44institutetext: Institute for Astronomy, University
of Hawai‘i at Mānoa, 34 Ohia Ku Street, Pukalani, HI 96768, USA
55institutetext: Institute for Solar Physics, Dept. of Astronomy, Stockholm
University, AlbaNova University Centre, SE-106 91 Stockholm, Sweden

# Spatial resolution effects on the solar open flux estimates

I. Milić 1122 R. Centeno 33 X. Sun 44 M. Rempel J. de la Cruz Rodríguez 3355

(Received ; accepted )

###### Abstract

Context. Spectropolarimetric observations used to infer the solar magnetic
fields are obtained with a limited spatial resolution. The effects of this
limited resolution on the inference of the open flux over the observed region
have not been extensively studied.

Aims. We aim to characterize the biases that arise in the inference of the
mean flux density by performing an end-to-end study that involves the
generation of synthetic data, its interpretation (inversion), and a comparison
of the results with the original model.

Methods. We synthesized polarized spectra of the two magnetically sensitive
lines of neutral iron around 630 nm from a state-of-the-art numerical
simulation of the solar photosphere. We then performed data degradation to
simulate the effect of the telescope with a limited angular resolution and
interpreted (inverted) the data using a Milne-Eddington spectropolarimetric
inversion code. We then studied the dependence of the inferred parameters on
the telescope resolution.

Results. The results show a significant decrease in the mean magnetic flux
density —related to the open flux observed at the disk center— with decreasing
telescope resolution. The original net magnetic field flux is fully resolved
by a 1m telescope, but a 20 cm aperture telescope yields a 30% smaller value.
Even in the fully resolved case, the result is still biased due to the
corrugation of the photospheric surface.

Conclusions. Even the spatially averaged quantities, such as the open magnetic
flux in the observed region, are underestimated when the magnetic structures
are unresolved. The reason for this is the presence of nonlinearities in the
magnetic field inference process. This effect might have implications for the
modeling of large-scale solar magnetic fields; for example, those
corresponding to the coronal holes, or the polar magnetic fields, which are
relevant to our understanding of the solar cycle.

###### Key Words.:

Sun: photosphere Sun: magnetic fields

##  1 Introduction

The physical parameters in the solar photosphere are generally inferred from
the intensity and polarization of the observed light (i.e., the polarized
spectrum) at the wavelengths corresponding to spectral line transitions.
Arguably the most interesting parameter is the magnetic field vector, the
measurements of which are essential for our understanding of the active
regions, flux emergence, magnetic reconnection, and various other aspects of
solar activity and dynamics. Notably, the photospheric magnetic field is used
as a boundary condition for the so-called magnetic field extrapolation
techniques (Régnier 2013), which allow us to model the magnetic field of the
outer solar atmosphere and the heliosphere. Specifically, the extrapolations
of the magnetic fields from the regions that correspond to the coronal holes
into the heliosphere yield a significant mismatch with the values obtained
from in situ measurements (the open flux problem; see Linker et al. 2017). A
possible way to resolve this problem is related to the fact that our current
photospheric diagnostics underestimate the open magnetic flux.

Reasons for a mismatch between a “true” value, which is, strictly speaking,
never known to us, and the inferred one can be numerous. First, simplified
spectral line formation and polarization models used for magnetic field
diagnostics introduce systematic errors that are non-trivial to find and
eliminate. The limited spatial, spectral, and temporal resolution of our
observations biases the results of our diagnostics. Non-negligible photon
noise results in significantly different errors in the inference of line-of-
sight and transversal components of the field. Finally, even a perfect
inference yields two equally valid orientations of the magnetic field, with
different physical implications. Away from the disk center, this results in
different vectors in the local reference frame, different radial components of
the magnetic field, and, therefore, different open fluxes.

A complete assessment of all of these factors is only possible if we simulate
the whole measurement process and compare the output to the input values of
the parameters used to calculate the synthetic observations. Leka & Barnes
(2012) performed such a study using examples of synthetic and observed data
and found differences between the original and inferred mean unsigned flux
density and the field inclination. Furthermore, Danilovic et al. (2016a) and
Danilovic et al. (2016b) studied the effect of telescope PSF straylight in the
retrieved inter-network magnetic field vector by means of 2D spatially coupled
inversions of Hinode data. These latter authors showed that the mean magnetic
field at an optical depth of unity is approximately 130 G. At higher layers,
the field strength is lower and the field is more horizontal. Recently, an
end-to-end study was performed for the GONG instrument (Harvey et al. 1996)
and is described in a series of papers by Plowman & Berger (2020a, b, c), who
pointed out numerous biases in the measurement process. Studies like these are
important because they allow us to modify the input models and to therefore
better understand the inference biases.

In the following analysis, we focus on the effects of the limited spatial
(i.e., angular) resolution of the telescope on the magnetic field inference at
the center of the solar disk. In this physical scenario, the line-of-sight
component of the field coincides with the radial direction. That is, our
inference (inversion) directly yields the radial component of the field. While
a poorer spatial resolution clearly results in a loss of small-scale details,
one would not expect a change in the net inferred radial magnetic field, that
is, averaged over the whole field of view (FoV). However, because the magnetic
field inference is a nonlinear process, we find the opposite outcome. In the
following section, we show how we synthesize the observational data and the
magnetic field inference. We follow up with section 3, where we describe our
magnetic field inversion and a comparison between the original and inferred
magnetic fields. Section 4 presents an illustration of this effect on the
real-life observed data, where SST/CRISP (Scharmer et al. 2003, 2008)
observations are used as the ground truth. In section 5, we present our
conclusions and plans for future work.

##  2 Data preparation and inversion

###  2.1 Synthetic spectra calculation

In this study, we use a state-of-the-art radiative-magnetohydrodynamic (RMHD)
simulation of the solar atmosphere performed with the MURaM code (Vögler et
al. 2005; Rempel 2014). Three-dimensional photospheric models obtained using
MURaM have been shown to accurately reproduce the observed properties of the
solar atmosphere (e.g., Danilovic et al. 2008). We use a single snapshot based
on a small-scale dynamo simulation presented in Rempel (2014), which considers
a computational box with horizontal extent of 24.576 $\times$ 24.576 Mm (1536
$\times$ 1536 cell point with 16 km spacing) and vertical extent of 8.192 Mm
(512 points with 16 km spacing). A uniform field of 30 G was added to the
small-scale dynamo solution with zero net flux and the simulation was evolved
for an additional 6 hours to allow for the formation of a magnetic network.
The last 15 minutes were evolved with nongray radiative transfer. In the
layers corresponding to the solar photosphere, the magnetic field forms a
complicated, plage-like magnetic structure (see Fig. 1) with patches of kG
fields. While the spatial ($x,y$) structure of the field is changing, the mean
vertical magnetic field is constant with height, that is, the vertical flux is
conserved as a consequence of periodic boundary conditions in the horizontal
directions. If we consider this atmosphere as a ground truth, an ideal
diagnostic method would retrieve a mean flux density equal to 30 G. This mean
flux density corresponds to an open flux of $\approx 1.8\times 10^{20}$ Mx. In
the context of this study, we refer to this flux as “open”, even though the
open flux is observationally typically defined over much larger areas.

Figure 1: Temperature at the photosphere ($\log\tau=0$), and the line-of-sight
magnetic field and velocity at ($\log\tau=-1$) in the simulated solar
photosphere used in this study.

We calculated the spectrum of the two magnetically sensitive neutral iron
lines around 630 nm using the SIR code (Ruiz Cobo & del Toro Iniesta 1992).
These are the spectral lines observed by the HINODE SOT/SP (Tsuneta et al.
2008) instrument and are a common choice for thermodynamic and magnetic
diagnostics of the solar photosphere. As we are primarily concerned with the
magnetic field, the synthesis is carried assuming local thermodynamic
equilibrium (LTE). Strictly speaking, iron is overionized in the photosphere
and precise modeling of the iron lines requires a non-LTE approach (e.g.,
Thévenin & Idiart 1999; Shchukina & Trujillo Bueno 2001; Smitha et al. 2021).

The calculation of the spectra yields a single polarized spectrum for each
pixel, sampled at the range from 630.1 to 630.3 nm with a 1 pm step size. For
a given model of the solar atmosphere, this is the spectrum we would measure
provided an instrument with infinite spatial (and spectral) resolution. When
observed with a telescope, spatial distribution of the incoming polarized
intensity gets convolved with the point spread function (PSF) of the optical
system, which is often approximated by the diffraction pattern of the entrance
aperture. We assume an ideal, Airy disk PSF (diffraction pattern of a circular
aperture) for four different diameters, namely 0.1, 0.2, 0.5, and 1.0 m, and
convolve synthetic monochromatic images of each Stokes parameter at each
wavelength with these PSFs to get four datasets, corresponding to different
spatial resolutions. For reference, a 1 m circular aperture at this wavelength
results in the resolution of 0.16 arcseconds, which corresponds to
approximately 120 km on the solar surface, or eight times larger than the
resolution pixel. Therefore, we perform a 2$\times$2 binning of the convolved
images to ease the manipulation and reduce inversion time. We note that the
Airy disk PSF alone is not enough to fully reproduce the spatial distribution
of the emitted radiation. Danilovic et al. (2008) demonstrated that additional
instrumental effects, such as defocus, binning, and so on, have to be
considered. Therefore, the effects described here would be slightly different
for examples of specific telescopes and instruments.

###  2.2 Inversions

Inversion is the process of fitting the observed Stokes spectrum with a
physically motivated model. The model parameters are the physical quantities
(in this specific case, magnetic field vector) that we want to infer from the
observed Stokes vector. The most realistic model is a fully stratified
atmosphere where temperature, pressure, magnetic field, and velocity are
allowed to freely vary with height. However, fitting such a model to the data
is a fairly difficult task (see the review by del Toro Iniesta & Ruiz Cobo
2016, for an in-depth description of spectropolarimetric inversions), and
simplifications are often needed. A robust trade-off between realism and
simplicity is the assumption of a Milne-Eddington atmosphere (M-E, e.g. del
Toro Iniesta 2003). In this case, spectral line formation is modeled by
several parameters that are constant with depth (magnetic field vector, line-
of-sight velocity, line strength, Doppler broadening, and damping) and a line
source function that varies linearly with optical depth. The M-E atmosphere
does not explicitly assume LTE, but it is most often used to model the LTE
lines formed in the photosphere, as there the source function can be
reasonably well approximated with a linear function. Codes based on the M-E
model are widely used in the data-inversion pipelines, such as VFISV (Borrero
et al. 2011), used for the inversion of SDO/HMI observations, MERLIN (Lites et
al. 2007), used for Hinode SOT/SP data, and MILOS (Orozco Suárez & Del Toro
Iniesta 2007), used for Solar Orbiter data.

Here, we chose a recently developed M-E inversion code,
PyMilne111https://github.com/jaimedelacruz/pyMilne (de la Cruz Rodríguez
2019). The code is compiled in C++, usable from Python, and makes full use of
OpenMP capabilities, thus enabling very fast inversion times even on personal
computers (hundreds to thousands of spectra per second). The use of a M-E
inversion scheme calls for a discussion on the vertical and horizontal
dependence of the magnetic field, that is, a consideration of the line-
formation height and the magnetic field filling factor.

The magnetic field in the solar atmosphere undoubtedly varies with depth, even
on scales comparable to spectral line formation regions. As the M-E model
assumes a constant magnetic field, it is not straightforward to relate the
inferred magnetic field vector to a specific atmospheric depth. A
comprehensive study by Borrero et al. (2014) showed that the M-E inversions
retrieve a weighted mean of the magnetic field over the line formation depths,
where the weighting function is the so-called response function (Beckers &
Milkey 1975). If we want to relate inferred magnetic field over an extended
FoV to a specific height, or even optical depth, we are introducing some bias
as we probe the field at different depths in different pixels. We show that
this effect leads to a disagreement in the inference of the net flux even in
the absence of the telescope PSF (section 3.2).

Another aspect is related to the limited angular resolution. In principle, the
magnetic field in the solar photosphere might be structured on very small
scales, that is, certainly smaller than the pixel size. The spectrum of the
observed pixel can then be modeled as a weighted combination of a magnetic and
nonmagnetic atmosphere, where the weighting is typically described by the so-
called filling factor. The nonmagnetic component is often referred to as stray
light even though it does not necessarily physically describe the stray light
in the telescope. Recently, we assessed the role of the filling factor in the
polar Hinode-like observations (Centeno et al. 2023) and found that M-E
inversions that use the filling factor still yield very large errors when
inferring intrinsic magnetic field properties. Additionally, the filling
factor can result in a severe overfitting of the observed spectra, and
potentially dubious interpretation of the results (e.g., pixels with very
small polarization signals can still yield strong magnetic fields). Therefore,
in the following analysis, we assume that the observed pixel is completely
permeated by the magnetic field, that is, that the filling factor is equal to
unity. We note that PyMilne allows the use of spatially coupled inversion
(discussed in de la Cruz Rodríguez 2019), which, in principle, can also
include the known telescope PSF (similar to van Noort 2012), which is a
feature we will explore in a follow-up paper to this one.

##  3 Inversion of synthetic observations

We inverted the original synthetic dataset (1536 $\times$ 1536 polarized
spectra), as well as the four degraded datasets (768 $\times$ 768 spectra
each) that correspond to telescope apertures of 1 m, 0.5 m, 0.2 m, and 0.1 m
(assuming ideal, clear-aperture telescopes). Each of the inversions yields a
single value for the magnetic field vector for each pixel. This allows us to
analyze the inferred vertical component of the magnetic field and therefore
the mean magnetic flux density. The latter multiplied by the surface area of
the FoV yields the open flux. We note that we use “inferred” here to emphasize
that these values come from the inversion of the polarized spectra, given the
assumed model, and are different from the true values.

Figure 2 shows the spatial distribution of the inferred line-of-sight magnetic
field for original data, and the spatial distributions corresponding to 1 m
aperture and 0.2 m aperture size. Visually, there is no significant difference
between the first two, while the third map shows significantly weaker and more
poorly resolved magnetic fields. On the other hand, the inferred mean flux
density for the three cases is equal to 32.8 G, 32.6 G, and 22.9 G,
respectively. Another quantity that is often analyzed is the mean unsigned
flux density, which for these three cases is equal to 48.9 G, 41.9 G, and 25.9
G, respectively.

Figure 2: Line-of-sight magnetic field as inferred from the synthetic
observations using Py Milne. Left: Original dataset. Middle: Dataset degraded
according to the PSF of a 1m telescope. Right: Same as the middle panel but
for a 0.2 m telescope. All three plots are in units of Gauss. The mean flux
density for these three cases is $32.8$ G, $32.6\,$G, and $22.9\,$G,
respectively.

It is clear that, due to the apparent cancellations, the decrease in the
telescope resolution causes us to see less magnetic field overall. This is
reflected in the decrease in the mean unsigned flux density with decreasing
spatial resolution, as discussed by, for example, Sánchez Almeida & Martínez
González (2011). In their well-known Figure 3, these latter authors compare a
large number of observational studies of the quiet Sun from multiple authors
and show that the mean unsigned flux density decreases monotonically with
decreasing spatial resolution. Here, the mean unsigned flux density is the
average value of $|B_{z}|$ over the FoV, where $B_{z}$ is the vertical
component of the magnetic field. The mean magnetic flux density, on the other
hand, is equal to the average of $B_{z}$, and, when multiplied by the area of
the FoV, gives us the total open flux. The open flux is proportional to the
number of magnetic field lines that extend vertically outside of the observed
region. We denote mean magnetic flux density with $\langle B_{z}\rangle$ and
mean unsigned flux density with $\langle|B_{z}|\rangle$.

While it is clear that the mean unsigned flux density must decrease with
telescope resolution because of the cancellation of the small-scale magnetic
features, intuitively we expect the mean flux density to be constant. The
reason for this is that, while poor telescope resolution might result in the
cancellation of polarization signals, we expect to see approximately the same
number of open field lines, even after the spatial averaging. Our results
(Fig. 3, top) reveal that this is not the case. $\langle B_{z}\rangle$ in the
original atmospheric cube is 30 G. The value inferred from the undegraded
synthetic spectra is slightly higher: 32.8 G. This value does not change if we
convolve our data with a PSF that corresponds to the 1m telescope (32.6 G).
However, convolution with the PSF of a 0.2m telescope yields a $30\%$ decrease
(22.9 G). The upper panel of figure 3 shows that, below the spatial resolution
of $\approx 100$ km (1m telescope), the estimated $\langle B_{z}\rangle$
quickly decreases down to almost half of its value, which is reached at the 10
cm aperture. For completeness, we also show (Fig. 3, bottom) the relationship
between the $\langle|B_{z}|\rangle$ and the telescope resolution
(corresponding to Fig. 3 of Sánchez Almeida & Martínez González 2011), in the
lower panel figure 3. We note that the latter is the log-log scale in order to
make it comparable with the results of Sánchez Almeida & Martínez González
(2011). In addition to the mean of the vertical magnetic flux density, the
whole inferred distribution significantly changes with the telescope
resolution. Figure 4 shows the changes in the inferred distribution of $B_{z}$
with the telescope size. While the distribution of data degraded with the 1m
telescope PSF only departs from the original distribution in the most extreme
values, the degradation with the 0.2m telescope PSF completely removes
magnetic fields stronger than 1 kiloGauss. For reference, the standard
deviations for the original and 1m cases are 179 and 159 Gauss, respectively,
while the application of the 0.2m PSF decreases this all the way down to 79
Gauss.

Figure 3: Effects of the telescope resolution on the average of the inferred
magnetic quantities. Top: The inferred mean flux density. The true value from
the model atmosphere is 30 Gauss. Bottom: Same, but for the mean unsigned flux
density. Figure 4: Distribution of the inferred magnetic flux density for
original synthetic observations and the observations degraded according to the
1m and the 0.2m telescope PSF.

###  3.1 Inversion nonlinearities

The reason for the decrease in the inferred mean flux density lies in the
nonlinearity of radiative transfer processes that relate the atmospheric
parameters to the emergent Stokes vector and, conversely, in the inversion
process. The Stokes vector $\hat{I}_{\lambda}(x,y)$ emerging from a 1D
atmosphere at the location $(x,y)$ is a result of a nonlinear mapping that
involves calculation of the emission and absorption terms, as well as the
solution of the polarized radiative transfer equation (see e.g., del Toro
Iniesta 2003). We denote this mapping by a functional $\mathcal{F}$:

$\hat{I}^{0}_{\lambda}(x,y)=\mathcal{F}[\@vec{B}(x,y,z),T(x,y,z),\@vec{v}(x,y,z)]=\mathcal{F}[\@vec{M}^{0}(x,y)].$
(1)

Hereafter, we use $\@vec{M}^{0}(x,y)$ to refer to all the physical parameters
relevant to the polarized line formation, absorbing their depth-dependence in
the vector $\@vec{M}^{0}$. In the detector, we measure the Stokes vector
convolved with the telescope PSF:

$\hat{I}_{\lambda}(x,y)=\int\int\hat{I}^{0}_{\lambda}(x^{\prime},y^{\prime})PSF(x^{\prime}-x,y^{\prime}-y)dx^{\prime}dy^{\prime},$
(2)

which translates to

$\hat{I}_{\lambda}(x,y)=\int\int\mathcal{F}[\@vec{M}^{0}(x,y)]PSF(x^{\prime}-x,y^{\prime}-y)dx^{\prime}dy^{\prime}.$
(3)

The estimated physical parameters are inferred (e.g., using an M-E inversion)
by performing an inverse of the functional $\mathcal{F}$:

$\@vec{M}(x,y)=\mathcal{F}^{-1}[\hat{I}_{\lambda}(x,y)].$ (4)

Therefore, the relationship between the measured physical parameters and the
original ones is

$\@vec{M}(x,y)=\mathcal{F}^{-1}[PSF\circledast\mathcal{F}[M^{0}(x,y)]],$ (5)

where $\circledast$ denotes convolution in the $x,y$ plane. While the
convolution is a linear operator, the radiative transfer and the inversion
process are not, and so the two will not commute. Therefore, we cannot say
that the inferred physical parameters in the presence of the telescope PSF are
simply the original parameters convolved by the telescope PSF. That is, a
spatially averaged magnetic field obtained from inversion is not necessarily
identical to the spatial average of the original magnetic field. This has
already been pointed out, for example, by Uitenbroek & Criscuoli (2011) for
the case of spectral line intensity, and recently by Plowman & Berger (2020b)
and Centeno et al. (2023) for the case of the photospheric magnetic field
vector. Additional biases are caused by the imperfections in the inverse
operator $\mathcal{F}^{-1}$ which cannot account for all of the complexities
in the forward model ${\mathcal{F}}$, and is often a very rough simplification
of reality.

In the case of simple photospheric magnetic field diagnostics, this bias can
be illustrated by considering a weak field approximation (WFA; e.g., del Toro
Iniesta 2003). Here, the circular polarization in the line is:

$V=kB_{\rm los}\frac{dI}{d\lambda},$ (6)

where the constant $k$ depends on the spectral line in question. For the disk-
center case: $B_{\rm los}=B_{z}$. ${dI}/{d\lambda}$ depends on the depth
gradient of the source function, which is generally smaller in intergranular
lanes than in granules. On the other hand, intergranular lanes typically
harbor stronger fields than granules. Considering four neighboring pixels,
that is, one magnetic intergranular pixel, and three nonmagnetic granular
pixels, and assuming that the intergranular pixel is permeated by $B_{z}=1000$
G field, but has a two-times smaller value of ${dI}/{d\lambda}$ than the
granular one: if we can resolve pixels individually and infer their magnetic
fields, we will estimate the mean flux density to be $\langle
B_{z}\rangle={1000{\rm G}}/{4}=250$ G. Now, if we cannot resolve individual
pixels, $V$ and $dI/d\lambda$ will follow from:

$\displaystyle\left(\frac{dI}{d\lambda}\right)_{g}$
$\displaystyle=2\left(\frac{dI}{d\lambda}\right)_{ig}=2x$ (7)
$\displaystyle\left(\frac{dI}{d\lambda}\right)_{\rm tot}$ $\displaystyle=7x$
(8) $\displaystyle V_{\rm tot}$ $\displaystyle=3V_{g}+V_{ig}=0+kB_{\rm
true}x,$ (9)

where we assumed that the intensity gradient in the intergranular lanes is
$x$, and that gradients can be added (as the wavelength derivative is a linear
operation). The inferred magnetic field from this unresolved case will now be:

$kBx=V_{\rm tot}=kB_{\rm unresolved}7x,$ (10)

where $B_{\rm true}$ is the magnetic field in the intergranular lane, equal to
1000 G, but $B_{\rm unresolved}={1000{\rm G}}/{7}\approx 143$ G, which is
almost two times less than in the spatially resolved case. The reason for this
discrepancy is again the nonlinearity of the inference method, which comes
into play even in this simple approach. The M-E approximation behaves
similarly to this case, because Stokes $V$ also depends linearly on the
gradient of the source function, which is different in different observed
pixels.

The effect of the telescope PSF on the estimated mean flux density has been
pointed out by Plowman & Berger (2020b), where the authors ascribe this effect
to the mixing of pixels with different velocities. To test the importance of
this effect, we performed two additional tests. First, we repeated the whole
experiment (spectral line synthesis and the M-E inversion for different levels
of PSF degradation) using the same MHD cube but setting all the velocities to
zero. This eliminates the line shifts and asymmetries altogether. The results
for the mean flux density we obtain by inverting this dataset are essentially
unchanged, suggesting that velocities play almost no role. Next, we set the
velocities to zero and made each atmosphere have identical temperature and
pressure stratification (but preserved the original magnetic field
stratification in each pixel). This results in a thermodynamically homogeneous
atmosphere in the $(x,y)$ plane, where pixels only differ in the magnetic
field structure. In this case, the convolution with the telescope PSF affects
the inferred mean magnetic flux density to a much lesser extent. The inversion
resulted in mean flux densities of 32 G for the case without spatial
degradation and 28.9 G for the PSF of a 10 cm telescope. The identical
temperature structure causes all pixels to have the same source function
gradient, thus eliminating the above effect to a high degree. Given that the
amplitudes of the Stokes $Q$, $U,$ and $V$ intensities are proportional to the
gradient of the source function with optical depth, this result is expected as
the spatial blurring has no net effect in this parameter. In the WFA, this
dependency is contained in the $dI_{\nu}/d\lambda$ term.

###  3.2 Depth dependence of the magnetic field

A less pronounced but evident effect is that the inferred mean magnetic flux
density —even from the ideal, spatially undegraded noise-free spectra— is
$\sim$10% higher than the original one. This has already been reported by
Schlichenmaier et al. (2023). Their explanation is that the more strongly
magnetic regions are also the less opaque ones, thus allowing us to see deeper
into the atmosphere, where the magnetic field is even stronger. We note that,
because of inhomogeneities in physical conditions in different pixels, we
effectively probe the magnetic field at different depths. That is, we are
inferring a magnetic field over a corrugated surface. To a first degree, this
surface can be approximated by a $\log\tau=$ const surface. In the case of the
Hinode lines, an estimate of the depth where the M-E inversion probes the
magnetic field is, on average, approximately $\log\tau=-1.5$ (Westendorp Plaza
et al. 1998; Centeno et al. 2023). To illustrate the effect of this
corrugation, in Fig. 5 we plot the mean magnetic flux density versus the
continuum optical depth in the MURaM simulation. The values range between 27 G
in the deep photosphere ($\log\tau=1)$ and 36 G in the upper-mid photosphere
($\log\tau=-2$). This amounts to around $30\%$ difference, which depends on
the opacity calculations used to convert geometrical height to optical depth.
We note that the vertically stratified inversions retrieve physical parameters
on an optical depth grid, and so according to Fig. 5, it is expected that
depth-dependent inversions will not result in the conservation of the mean
magnetic flux with optical depth. The only way to obtain a consistent solution
is to retrieve the true height stratification. This was mostly attempted
through post-processing of inversion results (see: Puschmann et al. 2010;
Löptien et al. 2018) but also by searching for a self-consistent MHD solution
(Riethmüller et al. 2017), and by using deep learning (Asensio Ramos & Díaz
Baso 2019). Recently, Borrero et al. (2019, 2021) demonstrated how to perform
this latter process self-consistently with the spectropolarimetric inversion,
but the method in its present form only works in highly magnetized
photospheric regions.

Figure 5: Variation of the mean magnetic flux density with optical depth in
the MURaM cube.

##  4 Test using SST/CRISP data

Our results in the previous section indicate that a decrease in the telescope
resolution results in a decrease in the estimated mean magnetic flux density
in the observed region. We expect this to depend on the magnetic field
distribution in the $(x,y)$ plane. To test our conclusions on real-life data,
we use very high-spatial-resolution observations of a plage close to the disk
center performed with the CRISP instrument (Scharmer et al. 2008) at the
Swedish Solar Telescope (SST, Scharmer et al. 2003). The data were processed
using the CRISPRED/SSTRED data pipeline (de la Cruz Rodríguez et al. 2015;
Löfdahl et al. 2021) and atmospheric seeing effects were compensated using the
Multi-Object-Multi-Frame-Blind-Deconvolution technique (Löfdahl 2002; Van
Noort et al. 2005). The observational setup is already described in Kianfar et
al. (2020), and so we summarize only the information relevant to our study.
The FoV is situated very close to the disk center (-135”, 77”, $\mu=0.99$) and
covers a 44 $\times$ 43 arcsec area, with a sampling of 0.059 arcseconds per
pixel. The Fe I 6301/6302 spectral lines are sampled using a nonequidistant
grid with a total of 16 wavelength points at 37 s cadence. For the purposes of
the experiment, we assumed that these observations fully resolve the field
structure. We then convolved it with an Airy disk PSF corresponding to a 20 cm
aperture, and inverted both data sets using PyMilne, taking into account the
CRISP spectral profile. We calculated the mean vertical flux density and found
it to be -80 G for the original CRISP data and -68 G for the degraded data
(Fig 6). The effect exists but is noticeably lower: 15%, compared to a 30%
loss of mean flux density found using the synthetic MURaM spectra. One
difference is that the structuring of the magnetic field in the plage is
different from the simulated magnetic field in the MURaM simulation. That is,
more of the flux is contained in the lower spatial frequencies, making it less
sensitive to the effects we find when analyzing the synthetic data. Another
possibility is that a part of the flux density is already missing due to the
PSF of the SST. That is, even high-resolution observations like these ones are
still a degraded version of the “real” Sun. Overall, 15% is a non-negligible
amount and is probably comparable to or larger than the standard uncertainties
in the magnetic field estimates.

Figure 6: LOS magnetic field from the original (left) and degraded (right) SST
observations. Both plots are in units of Gauss. The mean flux densities for
the two cases are $-80$ G and $-68\,$G, respectively.

##  5 Conclusions

The main conclusion of our work is that even an estimate of a spatially
averaged quantity —in this case, the mean magnetic flux density $\langle
B_{z}\rangle$— suffers from a bias when the telescope resolution is
insufficiently high. Here, we simulated this effect by calculating the spectra
of magnetically sensitive lines of neutral iron around 630 nm, applying
various levels of spatial PSF, and inverting the data using a conventional,
M-E model. This effect was previously found by Plowman & Berger (2020b), but
it was attributed to the presence of velocities. Here, we show that the most
important factor causing this bias is the nonlinearity in both the forward and
inverse model used to infer the magnetic field. Another reason for a
systematic error in the estimation of the mean flux density is the corrugation
of iso-optical depth surfaces compared to iso-height surfaces. Even though
these effects are rather simple to find and test for, the mechanisms causing
them lie in the physics of the spectral line formation in the inhomogeneous
atmosphere and are not trivial to correct. Synoptic observations, for example,
imply relatively low spatial resolution in order to cover the whole solar
disk. Coincidentally, these are the very same observations that are used to
compute the open flux and are likely to suffer from the biases explored in our
work. Linker et al. (2017) posed the open flux problem as a possible
misdiagnosis of the photospheric magnetic fields, which leads to a
disagreement between measured and extrapolated magnetic fields in the
heliosphere. The influence of spatial resolution on magnetic field
extrapolations has been demonstrated by, for example, DeRosa et al. (2015),
who degraded Hinode/SOT data, and by Fleishman et al. (2017), who directly
rebinned magnetic fields from the numerical simulation. Our results directly
identify some of the biases and confirm that some of the open flux in the
photosphere might be lacking due to the limited telescope resolution.

Overall, our findings are in support of using high-resolution observations to
perform magnetic fields extrapolations (e.g., Vissers et al. 2022), and
calibrating the low-resolution observations to the higher resolution ones
(e.g., Sainz Dalda 2017). To obtain a more complete understanding of the
structure of the magnetic field, we must pursue and develop more sophisticated
inversion techniques (e.g., Borrero et al. 2019). We will follow up this study
with a more in-depth analysis of the biases found in the polar observations,
following the conclusions made in Centeno et al. (2023), but focusing on the
effects of PSF, noise, and disambiguation. A reliable quantitative inference
of the open flux is especially critical in the scope of the upcoming
observations of the solar poles by the Solar Orbiter mission.

###### Acknowledgements.

We thank Milan Gošić, Bryan Yamashiro, and Marian Martinéz Gonzalez for
fruitful discussions and ideas related to the research presented in this
paper. IM thanks the solar magnetism group at KIS for their remarks on the
manuscript. We thank an anonymous referee, whose comments helped improve the
manuscript. IM and XS acknowledge support from NASA HGIO award 80NSSC21K0736.
IM also acknowledges the International Space Science Institute (ISSI) in Bern,
through ISSI International Team project 497 (Advanced Three-Dimensional
Modeling of the Magnetic Field in Active Regions on the Sun). RC acknowledges
support from NASA LWS Award 80NSSC20K0217. This material is based upon work
supported by the National Center for Atmospheric Research, which is a major
facility sponsored by the National Science Foundation under Cooperative
Agreement No. 1852977. We would like to acknowledge high-performance computing
support from Cheyenne (doi:10.5065/D6RX99HX) provided by NCAR’s Computational
and Information Systems Laboratory, sponsored by the National Science
Foundation. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (SUNMAG, grant agreement 759548). The Swedish 1-m Solar Telescope is
operated on the island of La Palma by the Institute for Solar Physics of
Stockholm University in the Spanish Observatorio del Roque de los Muchachos of
the Instituto de Astrofísica de Canarias. The Institute for Solar Physics is
supported by a grant for research infrastructures of national importance from
the Swedish Research Council (registration number 2021-00169).

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