# Coronal Hole Detection and Open Magnetic Flux

Jon A. Linker Predictive Science Inc., 9990 Mesa Rim Road, Suite 170, San
Diego, CA 92121, USA  Stephan G. Heinemann Max-Planck-Institut für
Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
Institute of Physics, University of Graz, Universitätsplatz 5, 8010 Graz,
Austria  Manuela Temmer Institute of Physics, University of Graz,
Universitätsplatz 5, 8010 Graz, Austria  Mathew J. Owens Space and Atmospheric
Electricity Group, Department of Meteorology, University of Reading, Earley
Gate, P.O. Box 243, Reading RG6 6BB, UK  Ronald M. Caplan Predictive Science
Inc., 9990 Mesa Rim Road, Suite 170, San Diego, CA 92121, USA  Charles N. Arge
Heliophysics Science Division, NASA Goddard Space Flight Center, Code 671,
Greenbelt, MD, 20771, USA  Eleanna Asvestari Department of Physics, University
of Helsinki, P.O. Box 64, 00014, Helsinki, Finland  Veronique Delouille Royal
Observatory of Belgium, Avenue Circulaire 3, B-1180 Bruxelles, Belgium  Cooper
Downs Predictive Science Inc., 9990 Mesa Rim Road, Suite 170, San Diego, CA
92121, USA  Stefan J. Hofmeister Institute of Physics, University of Graz,
Universitätsplatz 5, 8010 Graz, Austria  Columbia Astrophysics Laboratory,
Columbia University, 550 West 120th Street, New York, NY 10027, USA  Immanuel
C. Jebaraj Royal Observatory of Belgium, Avenue Circulaire 3, B-1180
Bruxelles, Belgium  Centre for mathematical Plasma Astrophysics, Department of
Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium  Maria S.
Madjarska Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg
3, 37077 Göttingen, Germany  Rui F. Pinto LDE3, DAp/AIM, CEA Saclay, 91191
Gif-sur-Yvette, France  IRAP, Université de Toulouse; UPS-OMP, CNRS; 9 Av.
colonel Roche, BP 44346, F-31028 Toulouse cedex 4, France  Jens Pomoell
Department of Physics, University of Helsinki, P.O. Box 64, 00014, Helsinki,
Finland  Evangelia Samara Royal Observatory of Belgium, Avenue Circulaire 3,
B-1180 Bruxelles, Belgium  Centre for mathematical Plasma Astrophysics,
Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven,
Belgium  Camilla Scolini Institute for the Study of Earth, Oceans, and Space,
University of New Hampshire, Durham, NH 03824, USA  University Corporation for
Atmospheric Research, Boulder, CO, USA  Bojan Vršnak  Hvar Observatory,
Faculty of Geodesy, University of Zagreb, Kačićeva 26, 10000, Zagreb, Croatia

(August 27, 2024)

###### Abstract

Many scientists use coronal hole (CH) detections to infer open magnetic flux.
Detection techniques differ in the areas that they assign as open, and may
obtain different values for the open magnetic flux. We characterize the
uncertainties of these methods, by applying six different detection methods to
deduce the area and open flux of a near-disk center CH observed on 9/19/2010,
and applying a single method to five different EUV filtergrams for this CH.
Open flux was calculated using five different magnetic maps. The standard
deviation (interpreted as the uncertainty) in the open flux estimate for this
CH $\approx$$26\%$. However, including the variability of different magnetic
data sources, this uncertainty almost doubles to $45\%$. We use two of the
methods to characterize the area and open flux for all CHs in this time
period. We find that the open flux is greatly underestimated compared to
values inferred from in-situ measurements (by 2.2-4 times). We also test our
detection techniques on simulated emission images from a thermodynamic MHD
model of the solar corona. We find that the methods overestimate the area and
open flux in the simulated CH, but the average error in the flux is only about
$7\%$. The full-Sun detections on the simulated corona underestimate the model
open flux, but by factors well below what is needed to account for the missing
flux in the observations. Under-detection of open flux in coronal holes likely
contributes to the recognized deficit in solar open flux, but is unlikely to
resolve it.

Sun: corona — Sun: magnetic fields — Sun: heliosphere — methods: numerical —
methods: data analysis

##  1 Introduction

The solar wind is a magnetized plasma that expands outward from the solar
corona to fill the interplanetary space. It plays a key role in heliophysics,
providing the medium by which solar-originating space weather-driving
phenomena, such as coronal mass ejections (CMEs) and solar energetic
particles, produce effects/impacts on Earth and on the surrounding space
environment. The solar wind is approximately structured into two types: slow
and fast with different sources (Schwenn et al., 1981). Fast solar wind
streams are associated with recurrent geomagnetic activity (Neupert & Pizzo,
1974) and are therefore of increased research interest. They have been
identified to originate from deep within coronal holes (Krieger et al., 1973),
where the predominantly open magnetic field allows plasma to escape easily
(Altschuler et al., 1972). Along these open magnetic field lines, the density
and temperature of the outflowing plasma falls rapidly with height, causing a
relatively low intensity emission of coronal holes (hereafter, CHs) in EUV and
X-Ray images, or correspondingly bright in He I 10830 absorption (Bohlin,
1977). The bulk of the Sun’s open magnetic flux that is measured in
interplanetary space is therefore expected to originate from CH regions.
However, recent investigations have shown that the open magnetic flux
identified in CHs underestimates the open magnetic flux in the heliosphere
deduced from in-situ measurements by a factor of two or more, referred to as
the “Open Flux Problem” (Linker et al., 2017; Lowder et al., 2017; Wallace et
al., 2019). While the fast wind is associated with the CHs themselves, the
more variable slow solar wind is associated with the CH boundaries. In
theories that invoke a quasi-static origin, the slow wind arises from regions
of large expansion factor near the boundaries (Wang & Sheeley, 1990; Cranmer
et al., 2007). Interchange reconnection (reconnection between open and closed
fields, Crooker et al., 2002) has been suggested as the source of a dynamic
slow solar wind (Fisk et al., 1998; Antiochos et al., 2011) and would most
easily occur near CH boundaries. Fisk & Kasper (2020) argue that recent
measurements from Parker Solar Probe (PSP, Fox et al., 2016) show that open
magnetic flux is transported by interchange reconnection. Identifying and
characterizing CHs and their boundaries is therefore crucial to understand the
origins of the solar wind and to assess the uncertainties in the
quantification of open magnetic flux.

The identification of CH regions is traditionally performed by visual
inspection of image data (Harvey & Recely, 2002). In recent years, several
automatic or semi-automatic routines have been developed for more objective
results (Henney & Harvey, 2005; Scholl & Habbal, 2008; Krista & Gallagher,
2009; Rotter et al., 2012; Lowder et al., 2014; Verbeeck et al., 2014;
Boucheron et al., 2016; Caplan et al., 2016; Garton et al., 2018; Heinemann et
al., 2019). In combination with photospheric magnetic field data, the
extracted CH area can be used to derive the open magnetic flux from that
region. If a detection method allows a full-Sun map of CHs to be created for a
given time period, then the solar open flux can be estimated entirely from
observations by overlaying the CH map onto a synchronic or diachronic (often
referred to as synoptic) magnetic map (Lowder et al., 2014; Linker et al.,
2017; Wallace et al., 2019).

As identified by Linker et al. (2017), there are two broad categories of
resolutions for this underestimate of the open flux: (1) Either the
observatory magnetic maps are underestimating the magnetic flux, or (2) a
significant portion of the open magnetic flux is not rooted in regions which
are dark in emission. Category (1) includes possible underestimates by the
magnetographs, which often disagree quantitatively (e.g., Riley et al., 2014),
or underestimates in specific regions, such as the poorly observed polar
regions (Riley et al., 2019). Category (2) raises the important question of
how well currently available CH detection methods perform and how they compare
to each other.

To address and resolve this issue, we formed an International Space Science
Institute (ISSI111http://www.issibern.ch/teams/magfluxsol/) team, and we
report the outcome of the first team meeting here. We study a well-observed
low-latitude CH and its associated Carrington Rotation (CR), that occurred
during solar minimum at the beginning of cycle 24 (CR2101, 2010-09-05 –
2010-10-03). We investigate the uncertainties in the calculation of open
magnetic flux from remote observations by exploring the variability in the
results that occur when different CH detection techniques, different
wavelengths, instruments, and different photospheric magnetic maps, are used.
As there is no “ground truth” measurement for the open flux on the Sun, we use
a thermodynamic MHD model (e.g., Mikić et al., 2018) to simulate the corona
for this time period and produce synthetic EUV emission images. The same
analysis that was performed on the observations is repeated for the simulated
data, where the “true” open flux is known. The observational and model results
are related to in-situ estimates of the heliospheric magnetic flux. From that
we asses the overall ability of detection methods to account for solar open
flux and identify potential sources of missing open flux.

##  2 Methodology and data

We focus on one particular CH observed on September 19, 2010, and the
surrounding time period (CR2101, 2010-09-05 – 2010-10-03). We selected this
time period and CH based on the following criteria i) availability of high-
resolution SDO/AIA and low noise SDO/HMI data, ii) a solar minimum time
period, when there is less solar activity and the coronal configuration is
simpler, iii) an isolated CH, i.e., not connected to a polar coronal hole nor
surrounded by strong active regions, with comparatively well-defined
boundaries at the solar surface for keeping projection effects to a lower
limit, iv) clear signatures of the associated high-speed stream from in-situ
data. Figure 1 shows the Sun on September 19, 2010 with the CH under study
located in the central part of the solar disk (panel a) as well as the related
solar wind high-speed stream at 1 AU from in-situ data (panel b).

Figure 1: (a) AIA/SDO 193Å full-disk image of the Sun on September 19, 2010.
The CH selected for this study is easily visible in the northern hemisphere.
(b) In-situ signatures of the associated solar wind (data provided by the OMNI
database), where the black line shows the solar wind bulk velocity, the green
line represents the plasma density and the blue line is the magnetic field
strength. The red-blue colored bars on the top represent the polarity of the
in-situ measured magnetic field calculated after Neugebauer et al. (2002),
with red being positive and blue negative polarity, and the time of the SDO
observation corresponds to the yellow vertical line.

###  2.1 CH Detection Methods

There are now several automated and semi-automated methods for detecting CH
boundaries from emission images, and these are often used to identify regions
of open magnetic flux. At the present time, the accuracy of these methods is
unclear, and there has been little intercomparison between the methods. The
uncertainty of these methods is therefore an open question that directly
impacts the larger question of why coronal estimates of open flux disagree
with inferences from in-situ measurements.

We apply six different but commonly used methods to this CH, and estimate the
uncertainties in CH detection, which in turn leads to uncertainties in the
observed open flux. The extraction methods used are: simple thresholding (THR;
Rotter et al., 2012; Krista & Gallagher, 2009), the Spatial Possibilistic
Clustering Algorithm (SPoCA, Verbeeck et al., 2014), Synchronic Coronal Hole
Mapping (PSI-SYNCH, Caplan et al., 2016), the Minimum Intensity Disk Merge
(PSI-MIDM, Caplan et al., 2019), the Coronal Hole Identification via Multi-
thermal Emission Recognition Algorithm (CHIMERA, Garton et al., 2018), and the
Collection of Analysis Tools for Coronal Holes (CATCH, Heinemann et al.,
2019). The CH area extraction methods are applied on high-resolution EUV data
in several wavelengths (171Å, 193Å, 211Å) from the Atmospheric Imaging
Assembly aboard the Solar Dynamics Observatory (AIA/SDO; Lemen et al., 2012).
The 193Å wavelength range is particularly favorable for the detection of CHs
due to the strong contrast between the low intensity CH region and brighter
surrounding quiet corona. In addition to AIA, we also use SWAP/PROBA2 174Å
(Seaton et al., 2013), XRT/HINODE (Golub et al., 2007), and 195Å data from the
EUVI instrument (Wuelser et al., 2004) aboard the STEREO spacecraft (Kaiser et
al., 2008). The different algorithms and methods are briefly described in the
following. Examples applying the different extraction methods are shown in
Figure 2.

Figure 2: Examples of the CH detection and extraction methods used in this
study applied on EUV image data. Images are for the CH of interest that
crossed the central meridian on September 19, 2010. Panels (a) through (f)
feature CATCH, CHIMERA, SPoCa, THR, PSI-SYNCH and PSI-MIDM respectively. To
focus on the CH of interest, full-sun PSI-SYNCH and PSI-MIDM detection maps
are cut out in a $\pm 90^{\circ}$ longitude vs. sine-latitude Stonyhurst
heliographic projection.

####  2.1.1 Simple Thresholding (THR)

Rotter et al. (2012) present a CH extraction method that applies simple
intensity thresholding, based on work described in Vršnak et al. (2007) and
Krista & Gallagher (2009). Following that approach, we use a threshold of 35%
of the median solar disk intensity to extract dark coronal features from
SDO/AIA 193Å images. The $35\%$ median intensity threshold has been found to
give consistent and reasonable results for CH boundaries, especially near the
maximum of solar cycle 24 (Hofmeister et al., 2017; Heinemann et al., 2019).

####  2.1.2 SPoCA

The SPoCA-CH-suite (Verbeeck et al., 2014) is a set of segmentation procedures
that allows decomposition of an EUV image into regions of similar intensity,
typically active regions, CHs, and quiet Sun. It relies on an iterative
clustering algorithm called fuzzy C-means, which minimizes the variance in
each cluster. Typically, the CH class corresponds to the class whose center
has the lowest pixel intensity value. The SDO Event Detection System runs the
SPoCA-suite to extract CH information from AIA images in the 193Å passband,
and uploads the entries every four hours to the Heliophysics Events
Knowledgebase (Hurlburt et al., 2012). The code for the SPoCA-suite is
available at https://github.com/bmampaey/SPoCA.

####  2.1.3 PSI-SYNCH & PSI-SYNOPTIC

The goal of the PSI-SYNCH algorithm (Caplan et al., 2016) is to create, as
accurately as possible, synchronic EUV and CH maps for the entire Sun. It was
originally developed and applied to the 2010-2014 time period when most or all
of the Sun was visible in EUV from the NASA STEREO and SDO spacecraft. All of
the PSI methods emphasize pre-processing of the image data. PSI-SYNCH applies
a point spread function (PSF) deconvolution to the full-disk images to remove
stray light, especially in the CHs, and produces non-linear limb brightening
correction factors and inter-instrument transformation factors using a running
one year average of disk data. The CH detection is applied to each disk image
using a dual-threshold region growing algorithm called EZSEG (image
segmentation code). After the detection, the results for each disk are merged
together into a single synchronic full-Sun CH map. Results, as well as the
open-source EUV pre-processing and EZSEG codes are made available at
http://www.predsci.com/chd. To produce EUV and CH maps that tend to be more
continuous for dark structures, each disk image is first mapped to its own
Carrington map, and then the three maps are merged by taking the lowest
intensity values of the overlap. For detection on the CH of interest, the maps
are cut out in a $\pm 90^{\circ}$ longitude vs. sine-latitude Stonyhurst
heliographic projection.

PSI-SYNCH provides full-Sun synchronic maps for those time periods when
combined SDO and STEREO images cover the entire Sun’s surface (2011-2014). In
September 2010, a portion of the Sun’s surface on the backside of the Sun was
not visible from the STEREO spacecraft. Therefore, combined SDO and STEREO
images cannot be used to generate a full-Sun map. To provide a full-Sun map
for CR2101 from which we can estimate the Sun’s total open flux, we use PSI-
SYNOPTIC. PSI-SYNOPTIC uses the same detection methodology, but instead
combines images over an entire solar rotation, with pixels weighted in
longitude by their proximity to disk center at the time of the observation.
This diachronic map uses only SDO AIA observations.

####  2.1.4 PSI-MIDM

A challenge for CH detection and area extraction based on single images stems
from the fact that EUV line-of-sight observations flatten the three-
dimensional structure in the low corona, which can cause nearby bright
structures to obstruct CHs. To mitigate this obstruction, an alternative
method to combine EUV disk images into a full-Sun map, referred to as the
Minimum Intensity Disk Merge (MIDM), is used. This builds on the PSI-SYNCH and
PSI-SYNOPTIC approaches, using an arbitrary number of vantage points and/or
images over time. PSI-MIDM (Caplan et al., 2019) uses the same EZSEG algorithm
as PSI-SYNCH for detection. Instead of using centrally weighted latitude
strips as in PSI-SYNOPTIC, MIDM takes full-disk images in the image- or time-
sequence and merges them based on which pixels in the overlap exhibit the
minimum intensity. This allows any part of the CH area observed from any
vantage point or time in the image sequence to be seen in the final map. This
can be performed even if only a single vantage point is available, by
combining images taken over time (i.e. SDO AIA observations). This creates a
trade-off between detecting rapid CH evolution versus revealing portions of
CHs obscured by bright loops (as can occur near active regions). For this
study, we created a PSI-MIDM map and detection using SDO AIA observations at a
6 hour cadence over all of CR 2101. As with PSI-SYNCH, a $\pm 90^{\circ}$
longitude vs. sine-latitude Stonyhurst heliographic projection is used to
focus on the selected CH.

####  2.1.5 CHIMERA

CHIMERA (Garton et al., 2018) uses the three SDO/AIA passbands where CHs are
predominantly visible (171Å, 193Å, and 211Å) to segment dark structures. The
extraction is based on the ratios and magnitudes of the emission from each
passband. CHIMERA is an automated CH detection and extraction algorithm and
derives robust boundaries which are continuously presented at
solarmonitor.org.

####  2.1.6 CATCH

The recently developed CATCH algorithm (Heinemann et al., 2019) is a
threshold-based CH extraction method, which uses the intensity gradient along
the CH boundary to modulate the extraction threshold. By minimizing the change
in the extracted area between similar thresholds, a stable boundary can be
found. CATCH also provides uncertainty estimations for all parameters. Due to
its concept and set-up, CATCH can be applied to any intensity-based EUV
filtergram to extract low intensity regions on the solar surface. The tool is
publicly available at https://github.com/sgheinemann/CATCH, including a link
to the VizieR catalogue - a sample of more than 700 CHs, ready for statistical
analysis.

###  2.2 Open flux derivation at the Sun

Photospheric magnetic maps show significant variability, from both the
underlying measurements at different observatories and the method of map
preparation. To address and quantify this issue, we use five different
magnetic map products to calculate the open flux within the extracted CH
boundaries. We obtain an estimate of the open magnetic flux for each CH
detection by overlaying the extracted CH boundaries on a photospheric magnetic
map taken at approximately the same time as the emission images. We integrate
the signed magnetic flux in each boundary (and also obtain the signed average
magnetic field) for each detection, using synoptic maps of the photospheric
magnetic field from three different observatories - Michelson Doppler Imager
(MDI; Scherrer et al., 1995) aboard the Solar and Heliospheric Observatory
(SOHO; Domingo et al., 1995), Helioseismic and Magnetic Imager (HMI; Schou et
al., 2012) aboard SDO (720s-HMI), and the ground-based Global Oscillation
Network Group instruments (GONG; Harvey et al., 1996). As these are derived
from line-of-sight (LOS) magnetograms, the radial magnetic field ($B_{r}$) is
obtained under the frequently used assumption that the field is radial where
it is measured in the photosphere (Wang & Sheeley, 1992). Additionally, we
used magnetic maps generated with the Air Force Data Assimilative Photospheric
flux Transport (ADAPT) model (Arge et al., 2010; Hickmann et al., 2015) using
both HMI and GONG full-disk magnetograms as input, for a total of five
different magnetic flux inputs. We note that the ADAPT model multiplied the
HMI values by 1.35, and the GONG values by 1.85, prior to assimilation. All of
the magnetic data were formatted to the same projection and resolution as the
detected CH boundaries.

###  2.3 Derivation of the Heliospheric Magnetic Field

Spacecraft with in-situ instruments directly measure the heliospheric magnetic
field (HMF) at a single point in space. The unsigned magnetic flux threading a
heliocentric sphere with measurement radius, $r$, can therefore be estimated
by $\Phi_{r}=4\pi r^{2}|B_{R}|$, where $B_{R}$ is the radial component of the
HMF, if it is assumed that the single-point measurement of $|B_{R}|$ is
representative of all latitudes and longitudes. From near-Earth space,
longitudinal structure of $|B_{R}|$ can be measured by considering an entire
Carrington rotation period and assuming the corona and HMF do not evolve
significantly over this time interval. Latitudinal invariance in $|B_{R}|$
(scaled for heliocentric distance) was confirmed by the $Ulysses$ spacecraft
on all three of its orbits (Lockwood et al., 2004; Smith & Balogh, 2008). Thus
the assumption that single-point measurements of the HMF can be used to
estimate $\Phi_{r}$ appears to be valid. This has been demonstrated
empirically by Owens et al. (2008). However, there is an additional issue that
$\Phi_{1AU}$ may not be equal to the unsigned flux threading the solar source
surface, $\Phi_{SS}$, which is the typical definition of open solar flux
(OSF). If the HMF becomes folded or inverted within the heliosphere,
$\Phi_{1AU}>\Phi_{SS}$. As suprathermal electrons move anti-Sunward on a
global scale (Pilipp et al., 1987), sunward motion can be used to identify
times when the HMF is locally folded (Crooker et al., 2004). For calculating
the heliospheric magnetic field, we take in-situ plasma and magnetic field
measurements from the Advanced Composition Explorer (ACE; Stone et al., 1998)
and its Solar Wind Ion Composition Spectrometer (SWICS; Gloeckler et al.,
1998), Solar Wind Electron Proton Alpha Monitor (SWEPAM; McComas et al., 1998)
and Magnetometer Experiment (MAG; Smith et al., 1998). Two of the CH detection
methods (PSI-MIDM and PSI-SYNCH/SYNOPTIC) provide CH detection and open flux
estimates over the entire Sun’s surface. We employ these methods to estimate
the amount of open flux in CR2101, and compare these results with estimates of
the interplanetary magnetic flux for this time period.

###  2.4 Simulating the open flux at the Sun

While remote solar observations allow us to infer the open solar magnetic flux
in the solar corona, we are not able to measure it directly. In an alternative
approach, we employ a thermodynamic MHD model (e.g., Mikić et al., 2018) for
CR2101, and we create a sequence of simulated AIA images for the same view
point of the real spacecraft over the course of the rotation. We apply the CH
detection methods to the simulated data, and compare with the “true” open flux
(known from the model) to further assess the effectiveness of detection
methods in accounting for open flux. The thermodynamic MHD model is briefely
described in the Appendix.

##  3 Inferring Open Flux from Coronal and Heliospheric Observations

###  3.1 Coronal Hole Detection

Figure 3: Detection of CH boundaries using different methods: (a) The AIA/SDO
$193$Å filtergram. (b) The AIA/SDO $193$Å filtergram overlaid with the minimum
and maximum boundary constructed by overlaying the 6 individual boundaries.
(c) The extracted CH boundary from CATCH using AIA/SDO 193Å; CHIMERA using
AIA/SDO 171/193/211Å; PSI-SYNCH using AIA/SDO 193Å and EUVI/STEREO 195Å; SPoCA
using AIA/SDO 193Å; thresholding (with $35\%$ of the median solar disk
intensity); and PSI-MIDM using multiple AIA/SDO 193Å images. (d) Stacked map
of the 6 binary extracted CHs. Darker pixels represent more agreement amongst
the detections.

We performed six detections for the selected CH using CATCH, CHIMERA, PSI-
SYNCH, SPoCA, THR and PSI-MDIM. The detections were each done using their
native input data and projection. For inter-comparison, the extracted CH
boundaries were projected to Carrington longitude at $1^{\circ}$ per pixel and
heliographic sine-latitude at $\tfrac{1}{90}$ per pixel, and smoothed using
spherical morphological operators of size 3 (Heinemann et al., 2019). The
resulting maps contained equal area pixels of roughly $9.4\times 10^{7}$ km2.
The comparison between the extracted boundaries is shown in Figure 3. We find
that the average extracted area is $8.89\pm 2.35\times 10^{10}$ km2, with the
SPoCA extraction providing the smallest value ($4.90\times 10^{10}$ km2), and
the largest value ($11.94\times 10^{10}$ km2) obtained from the PSI-MIDM
extraction. The areas of the maximum and minimum extractions differ by a
factor of $>2$ and the uncertainty, estimated as the standard deviation of the
mean of all CH areas, is roughly $\sigma_{\rm A,d}\approx 26\%$.

Figure 4: CH boundaries extracted using CATCH on different input EUV
filtergrams. The panels show: (a) The AIA/SDO $193$Å filtergram. (b) The
AIA/SDO $193$Å filtergram overlaid with the minimum and maximum boundaries
constructed by overlaying the 5 individual boundaries. (c) The different
boundaries extracted by the means of CATCH using input data from AIA/SDO 193Å
and 211Å, EIT/SOHO 195Å, SWAP/PROBA2 174Å and XRT/HINODE. (d) Stacked map of
the 5 binary extractions (where darker pixels represent an agreement of more
extractions).

To further explore the uncertainties in CH detection, we investigated how the
extracted CH boundary varied for different wavelengths and instruments. To
accomplish this task, we used CATCH, which employs a detection methodology
that performed close to the mean value of all the methods (cf. Table 1) and
moreover, is easily applicable to all intensity-based images. We applied CATCH
to five sets of input data (193Å AIA/SDO, 211Å AIA/SDO, 195Å EIT/SOHO, 174Å
SWAP/PROBA2 and XRT/HINODE). The comparison between the extracted boundaries
is shown in Figure 4. We find the average detected CH area to be $7.83\pm
2.00\times 10^{10}$ km2 for the five different data sets. The values range
from $10.23\times 10^{10}$ km2 (from EIT 195Å data) to $4.77\times 10^{10}$
km2 (from 174Å SWAP data) which is about a factor of two. The uncertainty here
is roughly in the same order as when using different extraction methods of
about $\sigma_{\rm A,\textsc{catch}}\approx 26\%$ of the mean. The low value
obtained with the 174Å wavelength should be viewed with some caution. This
line forms at a lower temperature range and at lower coronal heights than the
other lines, and therefore may image different structures in the CH.

Figure 5: The CH under study (a) shown in EUV and (b–f) in the different
photospheric magnetograms. The minimal and maximum CH boundaries are overlaid
as blue and red contours, respectively (cf. Figure 3). The magnetograms are
scaled to $\pm 50$ G.

To calculate the open flux within the extracted CH boundaries, we employ five
different photospheric magnetic maps (section 2.2). Figure 5 shows the five
different magnetic maps (MDI, HMI, GONG, GONG-ADAPT, HMI-ADAPT) overlaid with
the minimum and maximum stacked CH boundaries.

When applied to an individual magnetic map, we find that the variations in the
signed mean magnetic field density are rather small ($\sigma_{\rm{B}i}<9\%$)
between different extracted boundaries (varying both the detection method and
the input data). However, large deviations are derived between the different
magnetic maps with a mean magnetic field density of $-2.78\pm 0.67$ G within a
range of $-2.0$ to $-3.5$ G. This is equivalent to an uncertainty of
$\sigma_{\rm B}\approx 24\%$. When calculating the open flux from the CH area,
$A$, and underlying magnetic field, $B$, as $\Phi=A\times B$ for each
extraction and each map, we find an average of $(-23.59\pm 10.75)\times
10^{20}$ Mx ($\sigma_{\Phi,d}\approx 46\%$) for the different CH extraction
methods and $(-22.35\pm 9.53)\times 10^{20}$ Mx
($\sigma_{\Phi,\textsc{catch}}\approx 43\%$) for the different input data with
one extraction method. The values found range from $-10.9\times 10^{20}$ Mx to
$-35.32\times 10^{20}$ Mx between all CH extractions.

Table 1 summarizes the CH properties using different extraction methods,
magnetic field maps and native input data. Table 2 lists the CH properties
using CATCH with different input data and magnetic field maps. We see that the
ADAPT maps (using either GONG or HMI magnetograms) generally provide the
highest estimates of the mean magnetic field density and magnetic flux. This
is likely due to the multiplication factors applied to the input magnetograms
during assimilation, which is performed in part to counter perceived
underestimates of the magnetic flux.

Table 1: CH parameters derived from different extraction methods (CATCH, CHIMERA, SPoCA, PSI-SYNCH, PSI-MIDM and THR): area, $A$, intensity in 193Å image data, $I_{\rm 193}$, location of the center of mass, CoM (longitude and latitude), mean magnetic field strength, $B$, and magnetic flux, $\Phi$, for each magnetic map (MDI, HMI, HMI ADAPT, GONG, GONG ADAPT). | CATCH | CHIMERA | SPoCA | PSI-SYNCH | PSI-MIDM | THR | Mean
---|---|---|---|---|---|---|---
A [$10^{10}$ km2] | $8.2$ | $10.2$ | $4.9$ | $9.3$ | $11.9$ | $8.8$ | $8.9\pm 2.3$
I193 [DN] | $52.2$ | $58.7$ | $46.4$ | $53.8$ | $65.8$ | $53.0$ | $55.0\pm 6.6$
CoM${}_{\mathrm{\textsc{lon}}}$ [∘] | $-7.0$ | $-6.3$ | $-9.0$ | $-7.5$ | $-7.8$ | $-6.9$ | $-7.4\pm 0.9$
CoM${}_{\mathrm{\textsc{lat}}}$ [∘] | $23.3$ | $21.6$ | $25.6$ | $23.2$ | $24.5$ | $23.0$ | $23.5\pm 1.4$
B${}_{\mathrm{\textsc{mdi}}}$ [G] | $-2.8$ | $-2.7$ | $-2.9$ | $-2.6$ | $-2.7$ | $-2.7$ | $-2.7\pm 0.1$
B${}_{\mathrm{\textsc{hmi}}}$ [G] | $-2.1$ | $-1.9$ | $-2.2$ | $-2.0$ | $-1.9$ | $-2.1$ | $-2.0\pm 0.1$
B${}_{\mathrm{\textsc{hmi\leavevmode\nobreak\ adapt}}}$ [G] | $-3.1$ | $-2.8$ | $-3.3$ | $-3.1$ | $-2.8$ | $-3.1$ | $-3.0\pm 0.2$
B${}_{\mathrm{\textsc{gong}}}$ [G] | $-2.4$ | $-2.2$ | $-2.5$ | $-2.3$ | $-2.2$ | $-2.3$ | $-2.3\pm 0.1$
B${}_{\mathrm{\textsc{gong\leavevmode\nobreak\ adapt}}}$ [G] | $-3.5$ | $-3.0$ | $-3.7$ | $-3.3$ | $-3.0$ | $-3.4$ | $-3.3\pm 0.3$
$\Phi_{\mathrm{\textsc{mdi}}}$ [$10^{20}$ Mx] | $-22.5$ | $-27.7$ | $-14.0$ | $-24.7$ | $-32.5$ | $-23.8$ | $-24.2\pm 6.1$
$\Phi_{\mathrm{\textsc{hmi}}}$ [$10^{20}$ Mx] | $-17.4$ | $-19.7$ | $-10.9$ | $-19.1$ | $-22.6$ | $-18.2$ | $-18.0\pm 3.9$
$\Phi_{\mathrm{\textsc{hmi\leavevmode\nobreak\ adapt}}}$ [$10^{20}$ Mx] | $-25.7$ | $-28.9$ | $-16.2$ | $-28.6$ | $-33.3$ | $-27.1$ | $-26.6\pm 5.7$
$\Phi_{\mathrm{\textsc{gong}}}$ [$10^{20}$ Mx] | $-19.6$ | $-22.2$ | $-12.4$ | $-21.3$ | $-26.0$ | $-20.6$ | $-20.4\pm 4.5$
$\Phi_{\mathrm{\textsc{gong\leavevmode\nobreak\ adapt}}}$ [$10^{20}$ Mx] | $-28.2$ | $-30.9$ | $-17.9$ | $-30.6$ | $-35.3$ | $-29.7$ | $-28.8\pm 5.8$
Table 2: CH parameters derived from CATCH extraction and using different EUV filtergrams: area, $A$, intensity in image data, $I_{\rm 193}$, center of mass, CoM (latitude and longitude), mean magnetic field strength, $B$, and magnetic flux, $\Phi$, for each magnetic map (MDI, HMI, HMI ADAPT, GONG, GONG ADAPT).  | AIA 193Å | AIA 211Å | EIT 195Å | SWAP 174Å | XRT | Mean
---|---|---|---|---|---|---
A [$10^{10}$ km2] | $8.2$ | $7.4$ | $10.2$ | $4.8$ | $8.6$ | $7.8\pm 2.0$
I193 [DN] | $52.2$ | $51.8$ | $58.5$ | $47.2$ | $52.4$ | $52.4\pm 4.0$
CoM${}_{\mathrm{\textsc{lon}}}$ [∘] | $-7.0$ | $-8.1$ | $-8.0$ | $-10.4$ | $-7.7$ | $-8.3\pm 1.3$
CoM${}_{\mathrm{\textsc{lat}}}$ [∘] | $23.3$ | $24.2$ | $23.2$ | $26.3$ | $23.8$ | $24.2\pm 1.3$
B${}_{\mathrm{\textsc{mdi}}}$ [G] | $-2.8$ | $-2.6$ | $-3.1$ | $-3.0$ | $-2.9$ | $-2.9\pm 0.2$
B${}_{\mathrm{\textsc{hmi}}}$ [G] | $-2.1$ | $-1.9$ | $-2.3$ | $-2.4$ | $-2.3$ | $-2.2\pm 0.2$
B${}_{\mathrm{\textsc{hmi\leavevmode\nobreak\ adapt}}}$ [G] | $-3.1$ | $-3.2$ | $-3.1$ | $-3.5$ | $-3.2$ | $-3.2\pm 0.2$
B${}_{\mathrm{\textsc{gong}}}$ [G] | $-2.4$ | $-2.4$ | $-2.4$ | $-2.8$ | $-2.5$ | $-2.5\pm 0.2$
B${}_{\mathrm{\textsc{gong\leavevmode\nobreak\ adapt}}}$ [G] | $-3.5$ | $-3.6$ | $-3.4$ | $-3.9$ | $-3.5$ | $-3.6\pm 0.2$
$\Phi_{\mathrm{\textsc{mdi}}}$ [$10^{20}$ Mx] | $-22.5$ | $-19.4$ | $-31.4$ | $-14.2$ | $-25.2$ | $-22.5\pm 6.4$
$\Phi_{\mathrm{\textsc{hmi}}}$ [$10^{20}$ Mx] | $-17.4$ | $-14.3$ | $-23.4$ | $-11.5$ | $-19.6$ | $-17.2\pm 4.6$
$\Phi_{\mathrm{\textsc{hmi\leavevmode\nobreak\ adapt}}}$ [$10^{20}$ Mx] | $-25.7$ | $-23.6$ | $-31.9$ | $-16.8$ | $-27.9$ | $-25.2\pm 5.6$
$\Phi_{\mathrm{\textsc{gong}}}$ [$10^{20}$ Mx] | $-19.6$ | $-18.0$ | $-24.4$ | $-13.2$ | $-21.2$ | $-19.3\pm 4.1$
$\Phi_{\mathrm{\textsc{gong\leavevmode\nobreak\ adapt}}}$ [$10^{20}$ Mx] | $-28.2$ | $-26.4$ | $-34.4$ | $-18.4$ | $-30.0$ | $-27.5\pm 5.9$

The uncertainty in the open flux from a CH can, in principle, be divided into
the uncertainty from the CH extraction ($\sigma_{\Phi i}\approx 26\%$; cf.
Table 1 & Table 2) and the differences/uncertainties between different
magnetograms. From this we can conclude that for a typical extraction method
the uncertainty in the open flux derivation on a well-observed CH is
$\sigma_{\rm\Phi}\approx 43-46\%$.

###  3.2 Comparison with heliospheric open flux

Figure 6: In-situ observations during CR2101 from the ACE spacecraft. (a)
Pitch angle distribution of 272 eV electrons, with the intensity normalised at
each time step. (b) $\phi_{B}$, the HMF angle in the ecliptic plane. Red/blue
shaded regions show HMF pointed away/toward the Sun, while the blue/red lines
indicate the ideal Parker spiral directions for away/toward polarity HMF. (c)
$\theta_{B}$, the angle of the HMF out of the ecliptic plane. (d) $B_{R}$ the
radial HMF component. (e) $|B|$, the HMF intensity. (f) $V$, the solar wind
speed. (g) $n_{P}$, the proton density. Green vertical bars show intervals of
inverted HMF and yellow show intervals of undetermined HMF topology. (1-hour
means are shown.)

Figure 6 shows 1-hour means of ACE solar wind magnetic field and plasma data.
For determining the topology of the HMF over CR2101 we used 64-second
suprathermal electron and magnetic field data, following the methodology of
Owens et al. (2017). Accounting for inverted HMF and single-point sampling
uncertainty we derive for the heliospheric open solar flux $\Phi_{SS}$ in the
range $449$ to $559\times 10^{20}$ Mx, with a most probable value of
$482\times 10^{20}$ Mx. Another way of roughly estimating the heliospheric
open flux is to use simple averages (an hourly average and a daily average) of
the in-situ $B_{R}$, and then obtain the average of these over the entire
rotation. For the hourly averaged data, we obtain an estimate of $626\times
10^{20}$ Mx. This is almost certainly an overestimate of the open flux, for
the reasons described in section 2.3. For the daily average, we obtain
$464\times 10^{20}$ Mx. This likely underestimates the open flux, because for
these longer-time averages, $B_{R}$ is canceled in the vicinity of the
heliospheric current sheet as well as in regions with folded flux. These two
estimates bracket our most probable value ($482\times 10^{20}$ Mx) obtained
from the more detailed analysis.

Figure 7: (a) Full-Sun EUV AIA 193Å map and CH detections using PSI-MIDM. (b)
The same as (a) using PSI-SYNOPTIC.

Comparison of the open flux in CHs at the Sun with interplanetary measurements
requires CH detection for the entire solar surface. The PSI detection
techniques, PSI-SYNCH/SYNOPTIC and PSI-MIDM (see section 2.1), are designed to
produce a full-Sun map of EUV and extracted CHs. PSI-SYNCH cannot be used in
this capacity for CR2101, because at that time the combined view of STEREO-
EUVI and SDO-AIA did not extend over the entire Sun. PSI-MIDM uses multiple
views from AIA, and provides a CH map of the entire Sun over CR2101 shown in
Figure 7(a). We also employ PSI-SYNOPTIC, which is similar to PSI-SYNCH, but
is built up over a solar rotation in order to obtain a full-Sun view (Figure
7(b)). (We note that a portion of the south polar region of the Sun was not
visible during this period, and we assume that this is open. This lack of
visibility likely has a small impact on the estimate of open flux, as
described in the Appendix.)

To estimate magnetic flux from the global Sun CH maps, we employ three
synoptic maps, namely from HMI, MDI, and GONG, as they are built up over the
course of a solar rotation (ADAPT maps are not appropriate as they provide a
synchronic representation). Using the two full-Sun maps, PSI-MIDM and PSI-
SYNOPTIC, we sum the area of all the detected open field regions, and we sum
the fluxes in each CH individually, then take the absolute value of each
before summing them together. For PSI-MIDM, we obtain an estimate of
$84.0\times 10^{10}$ km2 for the open field area and $190\times 10^{20}$ Mx
for the open flux, using the HMI synoptic map. With the MDI synoptic map, the
open flux is $216\times 10^{20}$ Mx, and $160\times 10^{20}$ Mx for GONG. All
of these values are significantly less than the inferred interplanetary flux
of $482\times 10^{20}$ Mx. The open flux values are even smaller for PSI-
SYNOPTIC. With a detected area of $67.5\times 10^{10}$ km2, the open fluxes
are $154\times 10^{20}$ Mx (HMI), $174\times 10^{20}$ Mx (MDI), and $122\times
10^{20}$Mx (GONG).

When compared to all the other detection methods and input emission data, PSI-
MIDM provides the largest area and flux estimates for our selected CH. Yet the
estimate of the solar open flux from all detected CHs in CR2101 with PSI-MIDM
is well below the interplanetary flux estimate (by a factor of $\approx 2.2$)
in the best-case scenario. Our results imply that detected CHs over the entire
Sun, regardless of the detection method, contain significantly less flux than
is implied by interplanetary observations. However, there may be greater
uncertainty in the detection of CHs in other solar regions, particularly at
the Sun’s poles. Another way to test CH detection techniques is to apply them
to a model where the true answer is known. We describe this approach in the
following section.

##  4 Detection applied to Simulated Emission

In section 3, we found that the uncertainty estimates for CH area and open
flux for our well-observed CH are far below the amount required to account for
the large difference between our coronal and heliospheric open flux estimates
for the entire Sun during this time period. However, there is no “ground
truth” measurement for the open flux in a CH. There could be systematic errors
related to the geometry and properties of CHs for all detection methods. From
observational data alone, it is difficult to assess how (i) obscuration by
nearby overlying loops, (ii) foreshortening of the emission away from disk
center, especially near the poles, and (iii) variation in emission intensity
over the disk, affect detection of open magnetic flux. Thermodynamic MHD
models (e.g., Lionello et al., 2009; Downs et al., 2010, 2013; van der Holst
et al., 2014; Mikić et al., 2018; Réville et al., 2020) incorporate a
realistic energy equation that accounts for anisotropic thermal conduction,
optically thin radiative losses, and coronal heating, allowing the plasma
density and temperature to be computed with sufficient accuracy to simulate
EUV and soft X-ray emission observed from space. To assess how well detection
techniques perform when the true answer is known, we developed an MHD
simulation of CR2101 using the Magnetohydrodynamic Algorithm outside a Sphere
(MAS) code. The solution parameters closely resemble those used for the
coronal prediction for the August 21, 2017 total solar eclipse (Mikić et al.,
2018); a brief description of these parameters and the computation of
simulated emission is described in the Appendix. We describe the features of
the simulation relevant to our CH detection tests in the following section.

###  4.1 Properties of the Simulated Corona

Figure 8 shows some global diagnostics from the model results. Figure 8(a)
shows the surface magnetic map ($B_{r}$) used as the boundary condition. Red
indicates positive (outward) polarity ($+B_{r}$) and blue indicates negative
(inward) polarity ($-B_{r}$). Figure 8(b) shows a map of the simulated AIA
193Å emission from the model, with contours of the open field regions (red for
$+B_{r}$, blue for $-B_{r}$) overlaid on the map. A map of these open/closed
field regions is provided in Figure 8(c). These are the “target” areas for our
detection methods. Figure 8(d) shows $B_{r}$ overlaid on the open field map.
This is the “true” open magnetic flux in the model, the target that our
detection methods seek to extract. Figure 8(b) shows that in the model, in
addition to open field regions associated with dark emission and more unipolar
magnetic fluxes, there are also dark regions and open flux next to active
regions. This is in contrast to the observations (Figure 7), where this dark
emission is less apparent near the active regions, but may be obscured by
bright active region loops. Figures 8(e) and (f) show full-Sun CH detections
and are described in section 4.2.

Figure 8: Results from a thermodynamic MHD simulation of the solar corona for
CR2101.(a) Map of $B_{r}$ at the solar surface from an HMI synoptic map, the
boundary condition for the calculation. (b) Map of simulated AIA 193Å emission
from the model (see text). The open field regions of the model, calculated by
tracing field lines, are overlaid on the EUV map as colored contours. Red
colors indicate positive (outward) polarity, and blue colors indicate negative
(inward) polarity. (c) Map of the open (black) closed (white) field regions of
the model, (d) Map of $B_{r}$ overlaid on the open field map. This is the open
magnetic flux in the model. (e) CHs in the model (black) detected by PSI-MIDM.
(f) CHs in the model detected by PSI-SYNOPTIC.

To create simulated EUV images, we convolve the plasma density and temperature
from the model with the SDO/AIA instrument response functions. Synthetic
images are created by integrating the 3D volumetric emissivity along the line
of sight from a given viewpoint (see the Appendix for more details). The EUV
map in Figure 8(b) was constructed by integrating along radial lines of sight;
Linker et al. (2017) described a detection test with PSI-SYNOPTIC on a similar
EUV map. While this comparison yielded useful insights, in general, such a map
is more favorable for detection than real images from spacecraft instruments.

To provide more realistic conditions to test CH detection techniques, we
created a sequence of synthetic emission images in multiple wavelengths from
the MHD simulation as observed from the vantage point of SDO/AIA. Figure 9
shows an observational comparison for the date and time of our selected CH in
the different SDO/AIA filters. The comparison shows that the model has roughly
captured all prominent features of the corona at this time, including the
approximate location and size of active regions and CHs. However, the
simulated CHs are generally darker and more uniform than in real observations.
This is in part caused by the smoothness of the boundary map (Figure 8(a)),
which does not include the mixture of small scale parasitic polarities that
are prevalent at high resolution (compare with Figure 5). These small scale
structures likely contribute to the bright points of emission that tend to
“break up” CHs. This effect is likely to be especially prominent in the 171Å
line, where small-scale heating processes may dominate the lower temperature
plasma and exhibit more structure at smaller scale heights. The simulated
active regions also tend to be less structured than the observed ones.
Conversely, some of the simulated active region emission is over-bright
compared to the observations, and this may lead to more obscuration than in
observed structures. These less realistic attributes of the simulated corona
are related to resolution/computational cost and properties of the coronal
heating model. Further details are provided in the Appendix.

Figure 9: Comparison of observed and simulation AIA emission images on
September 19, 2010, at 17:59 UT. Top row: observed AIA emission in 171Å
(left), 193Å (middle), and 211Å (right). Bottom row: Simulated emission in the
same wavelengths.

###  4.2 Testing Detection Methods

Figure 10: CH boundaries extracted using different methods applied to
synthetic AIA/SDO emission, derived from the MHD model. (a) The synthetic
AIA/SDO $193$Å filtergram, with the contours of the true open field area shown
in cyan. (b) The same filtergram as (a), overlaid with true open field (cyan),
the minimum (dark blue) and maximum (red) boundary constructed by overlaying
the 5 individual boundaries. (c) The different boundaries extracted by the
means of CATCH (from AIA/SDO 193Å), CHIMERA (from AIA/SDO 171/193/211Å),
thresholding (with $35\%$ of the median solar disk intensity), SPoCA (AIA
193Å), and MIDM (AIA 193Å multiple images), along with the true open field.
(d) a stacked map of the 5 binary extractions (where darker pixels represent
an agreement of more extractions).

We applied our CH detection methods to the synthetic AIA images which we
created from the simulation, each method using its own processing methodology
as if this was a real observation. The results are shown in Figure 10. The
simulated emission in the selected CH (top left) shows less structure than the
real CH. The contours of the true open field regions reveal that magnetic
structure in the simulation is more complex. A closed field region is present
within the main body of the CH, but this feature is not revealed in emission,
possibly because of issues described in the Appendix. Contours of the
different detection schemes (Figure 10, bottom left) all incorporate this
region as open. The CATCH, SPoCA, and PSI-MIDM extracted boundaries are
similar, with CHIMERA and THR providing somewhat larger boundaries. The
minimal and maximal boundaries, constructed by overlaying the 5 individual
boundaries, along with the true open field boundaries, are shown in the top
right of Figure 10. A stacked map of the detections is shown in the bottom
right. To estimate the magnetic fluxes in the detections, we assume the values
in the boundary map that was used in the simulation (Figure 8(a)), i.e., we
assume there is no error in magnetic flux incorporated in the detections.

The numerical results for the detections, along with the true values, are
provided in Table 3. The average detected area of the CH ($11.5\times 10^{10}$
km2) is considerably larger than the true area ($7.4\times 10^{10}$ km2, but
the average strength of $B_{r}$ in the detected areas is smaller in absolute
value ($-2.1$ G) than the true value ($-3.1$ G). This occurs because regions
that were miss-identified as open by the detection methods had primarily mixed
polarity magnetic flux, lowering the magnitude of the average value. The
resulting average estimate for the open magnetic flux from this CH is
$24.2\times 10^{20}$ Mx for all the detections, only about 7% above the true
value. All but one of the methods overestimated the open magnetic flux in the
CH. The reasons for this are discussed further in section 5.3, but probably
occur because this is a relatively simple CH, situated away from any strong
active regions. We will see this is generally not the case when detecting open
magnetic flux from all CHs (i.e. over the full Sun).

Table 3: Extracted CH parameters: area ($A$), intensity in 193Å image data, (I193), location of the center of mass, (CoM, longitude and latitude), magnetic field strength ($B$), and magnetic flux ($\Phi$), from the synthetic 193 Å filtergrams using CATCH, CHIMERA, THR, SPoCA and PSI-MIDM. The magnetic field properties were derived using the input magnetic map for the MHD simulation.  | CATCH | CHIMERA | THR | SPoCA | PSI-MIDM | Mean | True
---|---|---|---|---|---|---|---
A [$10^{10}$ km2] | $10.5$ | $12.4$ | $15.5$ | $9.5$ | $9.7$ | $11.5\pm 2.5$ | $7.4$
I193 [DN] | $62.4$ | $66.2$ | $69.0$ | $60.0$ | $60.2$ | $63.6\pm 4.0$ | $61.2$
CoM${}_{\mathrm{\textsc{lon}}}$ [∘] | $-10.4$ | $-8.1$ | $-6.4$ | $-10.3$ | $-10.5$ | $-9.2\pm 1.8$ | $-9.8$
CoM${}_{\mathrm{\textsc{lat}}}$ [∘] | $26.6$ | $23.9$ | $21.2$ | $26.4$ | $26.5$ | $24.9\pm 2.4$ | $26.2$
B [G] | $-2.3$ | $-2.0$ | $-1.7$ | $-2.3$ | $-2.4$ | $-2.1\pm 0.3$ | $-3.1$
$\Phi$ [$10^{20}$ Mx] | $-24.0$ | $-25.3$ | $-26.6$ | $-21.6$ | $-23.2$ | $-24.2\pm 1.9$ | $-22.6$

To investigate how well CH detection methods can estimate the total open
magnetic flux, we again employ the PSI-MIDM and PSI-SYNOPTIC algorithms, as we
did for the observed CHs of CR2101. Figures 8(e) and (f) show the detected CHs
using these two methods. Comparison with Figures 8(c) and (d) shows that the
algorithms generally detect most of the true open field regions. The CH
structure identified by the methods is generally simplified compared to the
true open fields, and areas are overestimated in a number of regions,
especially the poles. Some smaller CHs are missed or not fully captured and as
these areas are associated with active regions, they contain a
disproportionate amount of open flux.

For the PSI-MIDM method, we find that the total Sun detected CH area is
$83.91\times 10^{10}$ km2 and the detected open flux is $285\times 10^{20}$
Mx. The PSI-SYNOPTIC method finds a detected CH area of $67.98\times 10^{10}$
km2 and detected open flux of $240\times 10^{20}$ Mx. The true open field area
was $62.71\times 10^{10}$ km2 and true open flux was $387\times 10^{20}$Mx.
Both methods overestimate the total areas of all CHs ($33.8\%$ by PSI-MIDM and
$8.4\%$ by PSI-SYNOPTIC) but underestimate the total open flux (PSI-MIDM
captures $73.7\%$ of the flux and PSI-SYNOPTIC captures $62.0\%$). These
underestimates are discussed further in the following section.

##  5 Discussion

###  5.1 CH Detection on a Well-Observed CH

Our comparison of detection methods for the area of the observed CH reveals a
standard deviation of $\sigma_{\rm A}\approx$ 26% from the mean value, which
we estimate to be the approximate uncertainty in the methods. The variability
in the magnetic fluxes from different magnetic field maps raises this
uncertainty to $\sigma_{\rm A}\approx\leavevmode\nobreak\ 43-46\%$. In the
methods comparison performed on the simulated CH, the standard deviation for
the detected areas was similar to the observed case, about $21\%$. However,
the mean of these values ($11.5\times 10^{10}$ km2) was actually $36\%$
greater than the true value ($7.4\times 10^{10}$ km2). All but one of the
methods overestimated the open flux in the simulated CH, but the standard
deviation in the open flux was much smaller ($8.4\%$) than for areas. The
actual error of the mean open flux compared with the true value was even less
($7\%$). This reflects the fact that if mixed polarity regions are mis-
identified as open, they don’t contribute as much to the flux estimate because
the opposite polarities cancel in the integration.

###  5.2 Detection of the Global Open Flux

Tables 4 and 5 summarize our full-Sun detections for both observations and the
model. The open flux inferred from heliospheric observations for this time
period was in the range $449-559\times 10^{20}$ Mx, with a most probable value
of $482\times 10^{20}$ Mx, corresponding to $B_{\rm R}=1.71$ nT at 1 AU. The
full-Sun detections, regardless of the magnetic map used, greatly
underestimate these values. The highest estimate ($0.77$ nT) comes from PSI-
MIDM with the MDI synoptic magnetic field map, and this detection was actually
an outlier for the detections on the selected individual CH. The uncertainties
we estimated for the area and open flux of the selected well-defined CH are
likely less than for full-Sun detections, where factors such as obscuration
and viewing geometry play a larger role. Our full-Sun detections on the model
corona at least partially account for these aspects, and indeed show that the
methods, while overestimating the CH area, actually underestimate the global
open flux (e.g., the true open flux is $35\%$ greater than the PSI-MIDM
estimate). However, even applying this factor to the PSI-MIDM full-Sun
detection of the observations leaves us well short of the estimated
heliospheric interplanetary flux.

Table 4: Areas and open fluxes derived from full-Sun detections of observed CHs for CR2101. Method | Magnetic Map | Area [$10^{10}$ km2] | Open Flux [$10^{20}$ Mx] | Heliospheric [$10^{20}$ Mx]
---|---|---|---|---
|  |  | ($B_{r}$ at 1 AU) | ($B_{r}$ at 1 AU)
MIDM | HMI | $84.0$ |  $191$ ($0.68$ nT) | $482$ ($1.71$ nT)
MDI |  $216$ ($0.77$ nT)
GONG |  $160$ ($0.57$ nT)
SYNOPTIC | HMI | $67.5$ |  $154$ ($0.55$ nT)
MDI |  $174$ ($0.62$ nT)
GONG |  $122$ ($0.43$ nT)
Table 5: Areas and open fluxes derived from full Sun detections of simulated CHs for CR2101. Method | Area [$10^{10}$ km2] | Open Flux [$10^{20}$ Mx] ($B_{r}$ at 1 AU) | True Area | True Flux
---|---|---|---|---
MIDM | $83.9$ |  $284$ ($1.01$ nT) | $62.7$ | $387$ ($1.37$ nT)
SYNOPTIC | $68.0$ |  $240$ ($0.85$ nT)

###  5.3 Overestimates of Open Flux in Individual CHs

There are two primary sources to the overestimate of the area and open flux on
the individual, simulated CH. The first can be seen by comparing the true open
field contour (cyan) in Figure 10 with the simulated emission and all of the
extracted CH boundaries in the figure. Pockets of closed field appear dark in
emission in the figure, and are indistinguishable from open field to the
detection methods. This may be less likely to occur on the real Sun, where
these small-scale loops may appear brighter in emission than they do in the
model. The absence of emission here may be due to deficiencies in the coronal
heating model for small-scale loops (see Appendix).

The second source of overestimation occurs because, as implemented here, the
methods do not account for the coronal height at which the bulk of the
emission begins to form in the EUV lines used in the detection (estimated to
be about $1.01$ R☉ for 193Å and 195Å emission, Caplan et al., 2016, see
section 4.2 and Figure 18). In general, the magnetic field expands with height
and the CH has a larger area at the height of detection than at its magnetic
source in the photosphere. Simply projecting the CH area downward on the
photosphere captures a larger area than the actual magnetic source. In the
model, the plasma at chromospheric temperature is artificially thick and the
193Å emission forms at $1.02$ R☉. Calculating the area of the true open field
at this height, we find that this rises to $9.3\times 10^{10}$ km2, much
closer to the detected areas (especially for SPoCA and PSI-MIDM). This result
suggests that CH detection methods may be able to improve the estimates of the
open magnetic flux by using a potential field model to extrapolate $B_{r}$ to
the height at which the emission forms, slightly lowering the flux estimate.

###  5.4 Underestimates of Total Open Flux

The full-Sun detections, when applied to the model corona, underestimate the
total open flux, with PSI-MIDM accounting for $73.7\%$ of the flux and PSI-
SYNOPTIC accounting for $62.0\%$. The reason for these underestimates can be
seen by comparing the true open field regions (Figures 8(c) and (d)) with the
detections (Figures 8(e) and (f)). There are several smaller-scale open field
regions that are under-detected or completely missed in the extracted CH
boundaries. These often are in the proximity of active regions, which contain
significant amounts of magnetic flux. For example, the two open field regions
near $270^{\circ}$ longitude and $45^{\circ}$ latitude are almost completely
missed in the extracted CH boundaries but account for $6\%$ of the total open
flux in the model. Linker et al. (2017) and Caplan et al. (2019) found that
open flux was underestimated by these detection methods for the same reasons.

###  5.5 Implications for the Open Flux Problem

Our comparisons of detection methods on our selected CH, for both the observed
and simulated cases, show reasonable agreement between the methods.
Uncertainty from the different magnetic map products contributes as much to
the variability as the detections themselves. The average error in the
detected open flux for the simulated CH was relatively small ($7\%$). The
full-Sun detections for the observed case all greatly underestimated the open
flux deduced from in-situ measurements, with the largest estimate still a
factor of 2.2 smaller than the interplanetary value. The schemes also
underestimated the open flux for the model corona. However, the model’s true
open flux was only $35\%$ greater than the estimate from the PSI-MIDM
detection.

The under-detection of open flux within the traditionally described CH areas
may well contribute to the open flux problem, however, it seems unlikely that
it is the only reason and, therefore, cannot resolve it.

We note that the open flux in the simulated corona was relatively close
($\approx 80\%$) to the in-situ value. However, the open field regions in the
model at mid-latitudes and near active regions are larger and more obvious
than in the observations. If regions like this exist on the real Sun and
contribute to the open flux, they would have to be considerably more obscured
than occurs in the model.

One possible resolution to the open flux problem is that the under-detection
of open flux (e.g., near active regions), in combination with systematic
underestimates of magnetic flux by magnetographs (either everywhere on the
Sun, or just at the poles), could account for the missing open flux. In this
regard, the behavior of CHs at the poles could be especially important, and
our present observational views of the Sun’s poles limit our ability to
resolve this question. The latter part of the Solar Orbiter mission, when the
spacecraft will reach latitudes of $\sim$30∘, may yield clues to the
importance of the polar contribution. Ultimately, a mission that fully images
the Sun’s poles (such as the Solaris mission, Hassler et al., 2019, 2020) can
resolve the contribution of the Sun’s polar regions to the open flux.

A second possibility is that a significant portion of the open flux is rooted
at the Sun, but continually undergoes interchange reconnection, and the
mixture of open and closed field lines are not obviously dark in emission.
While interchange reconnection has been advocated as an explanation for the
origin of the slow solar wind (e.g., Abbo et al., 2016, and references
therein), it is not clear what emission properties the plasma on these field
lines would possess. Therefore, with the present state of the theories/models,
it is difficult to either completely confirm or falsify this idea from
observations alone. An advanced model that simulates the time-dependent
evolution of the corona and demonstrates the observed emission properties
would seem to be necessary to progress beyond the present qualitative
arguments.

A third possibility is that the disparity between observed coronal and
heliospheric open flux is not related to solar observations, but to the
behavior of the interplanetary magnetic field. The discovery of long intervals
of “switchbacks” in the interplanetary magnetic field from PSP (Bale et al.,
2019; Kasper et al., 2019) suggests that folded flux could be more ubiquitous
than previously thought, and lead to increases in the magnitude of $B_{\rm R}$
measured in-situ at increasing distance (i.e., 1 AU) from the Sun (Macneil et
al., 2020). However, comparison of PFSS and MHD models with PSP observations
(Badman, S. et al., 2020; Riley, P. et al., 2021) suggest that the models
significantly underestimate the field strength even at the perihelion
distances that PSP has reached thus far, though more detailed accounting for
switchbacks may be necessary. Large amounts of disconnected flux in the
heliosphere could also account for the missing open flux. This has generally
been considered unlikely (Crooker & Pagel, 2008), but recent PSP observations
of reconnection in the heliospheric current sheet (Lavraud et al., 2020; Phan,
T. D. et al., 2020) indicate that this process appears to be more prevalent
than previously thought.

##  6 Summary

We have investigated CH detection techniques to characterize the uncertainty
in characterizing CH area and open flux from observational EUV data. Starting
from a well-observed, near-disk center CH, we applied six different detection
methods to deduce the area and open flux. We also applied a single method to
five different EUV filtergrams for this CH. Open flux was calculated for all
of the detections using five different magnetic maps. Using the standard
deviation as a measure of the uncertainty, we find that the uncertainty in the
estimate of open flux for this particular CH was $\approx 26\%$. When
including the variability in the different magnetic data sources, this
uncertainty rises to $43-46\%$. We used two of the methods to characterize the
area and open flux for all CHs during CR2101. We find that the open flux is
greatly underestimated compared to the value inferred from in-situ
measurements, by a factor of 2.2–4. As there is no “ground truth” measurement
of open flux in CHs, we tested our detection techniques on simulated emission
images from a thermodynamic MHD model of the solar corona for this time
period, where the true values of the model are known. The variability in the
detected area of the simulated CH is similar to the observed case. The methods
generally overestimate the area and open flux in the simulated CH, but the
average error in the flux is only about $7\%$. The full-Sun detections on the
simulated corona underestimate the model open flux (by $35\%$ for PSI-MIDM),
but this factor is well below what is needed to account for the missing flux
in the observations. Our results imply that under-detection of open flux in
what are traditionally considered to be coronal holes likely contributes to
the open flux problem, but is unlikely to resolve it.

We thank the International Space Science Institute (ISSI, Bern) for the
generous support of the ISSI team “Magnetic open flux and solar wind
structuring in interplanetary space” (2019-2021). JAL, RMC, and CD were
supported by the NASA Heliophysics Guest Investigator program (grant
NNX17AB78G), the NASA HSR program (grant 80NSSC18K0101 & 80NSSC18K1129), AFOSR
(contract # FA9550-15-C-0001), the NSF PREEVENTS program (grant ICER1854790)),
and the STEREO SECCHI contract to NRL (under subcontract N00173-19-C-2003 to
PSI). Computational resources were provided by NASA’s NAS (Pleiades) and NSF’s
XSEDE (TACC & SDSC). BV acknowledges the financial support by the Croatian
Science Foundation under the project 7549 “Millimeter and submillimeter
observations of the solar chromosphere with ALMA.” SJH was supported, in part,
by the NASA Heliophysics Living With a Star Science Program under Grant No.
80NSSC20K0183. EA acknowledges support from the Academy of Finland
(Postdoctoral Researcher Grant 322455). MO is funded by part-funded by Science
and Technology Facilities Council (STFC) grant numbers ST/R000921/1 and
ST/V000497/1. CS acknowledges support from the Research Foundation – Flanders
(FWO, strategic base PhD fellowship No. 1S42817N), and from the NASA Living
With a Star Jack Eddy Postdoctoral Fellowship Program, administered by UCAR’s
Cooperative Programs for the Advancement of Earth System Science (CPAESS)
under award no. NNX16AK22G. ES acknowledges support from a PhD grant awarded
by the Royal Observatory of Belgium. JP acknowledges support from the SolMAG
project (ERC-COG 724391) funded by the European Research Council (ERC) in the
framework of the Horizon 2020 Research and Innovation Programme, and the
Finnish Centre of Excellence in Research of Sustainable Space (Academy of
Finland grant number 312390). ICJ acknowledges support from a PhD grant
awarded by the Royal Observatory of Belgium.

##  Appendix A Thermodynamic MHD model

We developed an MHD simulation of CR2101 using the Magnetohydrodynamic
Algorithm outside a Sphere (MAS) code. The method of solution, including the
boundary conditions, has been described previously (e.g., Mikic & Linker,
1994; Linker et al., 1999; Lionello et al., 1998, 2009). The solution
parameters closely resemble those used for the coronal prediction for the
August 21, 2017 total solar eclipse (Mikić et al., 2018), including a Wave-
Turbulence-Driven (WTD) model of coronal heating (Downs et al., 2016). The
simulation utilized a $288\times 327\times 699$ nonuniform $r,\theta,\phi$
mesh, with the smallest radial cells of $\Delta r=0.00043R_{S}$ ($\approx 300$
km) near the solar surface and $\Delta r=0.83R_{S}$ at the outer boundary of
$30R_{S}$. For the co-latitude mesh, $\Delta\theta=0.51^{\circ}$ near the
equator and $0.86^{\circ}$ near the poles, and for the longitudinal mesh,
$\Delta\phi=0.51^{\circ}$ was uniform. The boundary condition was derived from
the HMI LOS synoptic map for CR2101, available at
http://jsoc.stanford.edu/HMI/LOS_Synoptic_charts.html. As in Mikić et al.
(2018), we multiplied the HMI map by a factor of 1.4 to account for the
difference between HMI and MDI data (Liu et al., 2012), which was previously
used to calibrate the heat flux in the model. We divide out this 1.4 factor
prior to computing the true and estimated magnetic fluxes in the detected CHs
described in section 4.2, to make these more comparable with the values
derived from observations.

The instrument response functions were developed using the AIA v6 calibration,
the CHIANTI 8.0.2 database (Del Zanna et al., 2015) and the CHIANTI hybrid
abundances (Fludra & Schmelz, 1999), based off (Schmelz et al., 2012).
Synthetic images are created by integrating the 3D volumetric emissivity along
the line of sight from a given viewpoint. The EUV map in Figure 8(b) was
constructed by integrating along radial lines of sight; Linker et al. (2017)
described a detection test with PSI-SYNOPTIC on a similar EUV map. While this
comparison yielded useful insights, in general, such a map is more favorable
for detection than real images from spacecraft instruments. While obscuration
of CHs from bright loops can still occur, this line of sight occurs only at
disk center. Away from disk center (especially at higher latitudes) more
obscuration may occur, and polar regions are especially foreshortened.

To test CH detection techniques under more realistic conditions, with a
dataset more akin to those actually produced by AIA, we created a sequence of
synthetic emission images in multiple wavelengths from the MHD simulation as
observed from the vantage point of SDO/AIA. The B0 angle (i.e., the
heliographic latitude of the central point of the solar disk) of the Sun is
included in the geometry. A set of images was created approximately every six
hours, for a total of 111 image sets. As described in section 4.1, smoothing
of the magnetic map reduces the presence of small-scale, mixed polarities, and
these provide important contributions to the complexity of real emission
images. The amount of smoothing of the map is in turn determined by the
available resolution for the MHD simulation, which strongly influences the
computational cost. A second simplification of the model is the attempt to
describe all of coronal heating with the simplified WTD description (for
details, see Downs et al., 2016). The origin of coronal heating is, of course,
controversial. The WTD mechanism, even if proven generally valid, may not be
applicable to heating at all coronal scales, as small-scale heating may be
dominated by other mechanisms - this may be more important for the 171Å line,
that has contributions from lower temperature plasma and exhibits more
structure at smaller scale heights. This effect can only be explored by
performing much higher resolution simulations than the one we employ here.
Furthermore, to capture the solar atmosphere’s thin transition region while
still modeling the vast scales of the solar corona, the simulation
artificially broadens the transition region by modifying the thermal
conduction and radiative losses at lower temperatures. This approach (Lionello
et al., 2009; Mikić et al., 2013) has been shown to accurately reproduce
coronal solutions at higher temperatures (for this case, $\ >400,000$K) but
can significantly modify the density (and thus emission) at lower temperatures
- this effect is again most likely to influence simulated 171Å emission.
Accuracy at lower temperatures can be provided at the cost of smaller cells in
the transition region.

Despite the aforementioned shortcomings, the simulated emission images still
provide a robust test of CH detection techniques, including obscuration by
overlying structures, orders of magnitude differences in emission intensity
between different portions of the solar disk, and realistic geometry. As with
the observations, a small portion of the southern pole is not visible from our
simulated viewpoint. In our full-Sun detection tests, we assumed this region
was open, the same as we did in the observed case. This turns out to impact
the open flux estimate by less than 1%, compared to the true value of the
model. Therefore, we expect this assumption to also have minimal impact on the
full-Sun estimates for the observed case.

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