Coronal Hole Detection and Open Magnetic Flux

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; Krista & Gallagher 2009; Rotter et al. 2012), 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 (171, 193, 211 Å) from AIA aboard SDO (Lemen et al. 2012). The 193 Å 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 174 Å (Seaton et al. 2013), XRT/HINODE (Golub et al. 2007), and 195 Å 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 below. Examples applying the different extraction methods are shown in Figure 2.

Zoom In Zoom Out Reset image size

2.1.1. Simple Thresholding (THR)

2.1.2. SPoCA

2.1.3. PSI-SYNCH and PSI-SYNOPTIC

2.1.4. PSI-MIDM

2.1.5. CHIMERA

2.1.6. CATCH

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.

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 Φ_r = 4π r²∣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, the 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 Φ_r appears to be valid. This has been demonstrated empirically by Owens et al. (2008). However, there is an additional issue that Φ_{1 au} may not be equal to the unsigned flux threading the solar source surface, Φ_SS, which is the typical definition of open solar flux. If the HMF becomes folded or inverted within the heliosphere, Φ_{1 au} > Φ_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.

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 viewpoint 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 briefly described in the Appendix.

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 intercomparison, the extracted CH boundaries were projected to Carrington longitude at 1° 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 × 10⁷ km². The comparison between the extracted boundaries is shown in Figure 3. We find that the average extracted area is 8.89 ± 2.35 × 10¹⁰ km², with the SPoCA extraction providing the smallest value (4.90 × 10¹⁰ km²), and the largest value (11.94 × 10¹⁰ 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 σ_A,d ≈ 26%.

Zoom In Zoom Out Reset image size

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 of the methods (see Table 1) and moreover, is easily applicable to all intensity-based images. We applied CATCH to five sets of input data (193 Å AIA/SDO, 211 Å AIA/SDO, 195 Å EIT/SOHO, 174 Å 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 ± 2.00 × 10¹⁰ km² for the five different data sets. The values range from 10.23 × 10¹⁰ km² (from EIT 195 Å data) to 4.77 × 10¹⁰ km² (from 174 Å 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 σ_A,catch ≈ 26% of the mean. The low value obtained with the 174 Å 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.

Zoom In Zoom Out Reset image size

To calculate the open flux within the extracted CH boundaries, we employ five different photospheric magnetic maps (Section 2.2). To compute the open magnetic flux, we sum the magnetic field values from the selected magnetic map within the equal area pixels of the CH map (described at the beginning of this section). Therefore, the magnetic flux in a CH is computed at the same resolution for all magnetic maps, regardless of the resolution of the original map. We do not consider the noise levels of the different maps, but the signed magnetic flux is not likely to be sensitive to this quantity, as the noise will tend to cancel in the integral. For example, Caplan et al. (2021) recently showed that open magnetic flux is insensitive to resolution in Potential Field Source Surface (PFSS) models, over a wide range of scales. Figure 5 shows the five different magnetic maps (MDI, HMI, GONG, GONG-ADAPT, and HMI-ADAPT) overlaid with the minimum and maximum stacked CH boundaries.

Zoom In Zoom Out Reset image size

When applied to an individual magnetic map, we find that the variations in the signed mean magnetic field density are rather small (σ_Bi < 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 ± 0.67 G within a range of −2.0 to −3.5 G. This is equivalent to an uncertainty of σ_B ≈ 24%. When calculating the open flux from the CH area, A, and underlying magnetic field, B, as Φ = A × B for each extraction and each map, we find an average of (−23.59 ± 10.75) × 10²⁰ Mx (σ_Φ,d ≈ 46%) for the different CH extraction methods and (−22.35 ± 9.53) × 10²⁰ Mx (σ_Φ,catch ≈ 43%) for the different input data with one extraction method. The values found range from −10.9 × 10²⁰ Mx to −35.32 × 10²⁰ 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.

The uncertainty in the open flux from a CH can, in principle, be divided into the uncertainty from the CH extraction (σ_Φi ≈ 26%; see Tables 1 and 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 σ_Φ ≈ 43%–46%.

Figure 6 shows 1 hr means of ACE solar wind magnetic field and plasma data. For determining the topology of the HMF over CR2101, we used 64 s 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 Φ_SS in the range 449–559 × 10²⁰ Mx, with a most probable value of 482 × 10²⁰ 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 × 10²⁰ 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 × 10²⁰ 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 × 10²⁰ Mx) obtained from the more detailed analysis.

Zoom In Zoom Out Reset image size

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 due to the ∼7° tilt of the Sun's rotation axis toward Earth during this time period, a portion of the south polar region of the Sun was not visible, 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.)

Zoom In Zoom Out Reset image size

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 of 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. We compute the fluxes in the individual CHs of the full-Sun map in a manner similar to that described in Section 3.1, but we integrated the fluxes on a higher-resolution (1600 × 3200) latitude–longitude grid, to ensure accuracy in the polar regions. For PSI-MIDM, we obtain an estimate of 84.0 × 10¹⁰ km² for the open field area and 190 × 10²⁰ Mx for the open flux, using the HMI synoptic map. With the synoptic map, the open flux is 216 × 10²⁰ Mx, and 160 × 10²⁰ Mx for GONG. All of these values are significantly less than the inferred interplanetary flux of 482 × 10²⁰ Mx. The open flux values are even smaller for PSI-SYNOPTIC. With a detected area of 67.5 × 10¹⁰ km², the open fluxes are 154 × 10²⁰ Mx (HMI), 174 × 10²⁰ Mx (MDI), and 122 × 10²⁰ Mx (GONG).

When compared to all of 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 ≈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.

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 193 Å 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)–(h) show full-Sun CH detections and are described in Section 4.2.

Zoom In Zoom Out Reset image size

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 LOS from a given viewpoint (see the Appendix for more details). The EUV map in Figure 8(b) was constructed by integrating along radial LOS; 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. This is especially true in the polar regions, where images obtained from ecliptic-based instruments suffer from considerable foreshortening.

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 171 Å 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.

Zoom In Zoom Out Reset image size

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 (shown in cyan) reveal that the magnetic structure in the simulation is more complex. Two closed field regions (cyan circles) are present within the main body of the CH, but these features are not revealed in emission (possible reasons for this are discussed in Section 5.3 and the Appendix). Contours of the different detection schemes (Figure 10, bottom left) all incorporate these regions 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 five 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.

Zoom In Zoom Out Reset image size

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 × 10¹⁰ km²) is considerably larger than the true area (7.4 × 10¹⁰ 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 misidentified 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 × 10²⁰ Mx for all of 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).

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 global EUV maps built up with these detection algorithms. The dark band near the southern pole in these images occurs because of the tilt of the Sun's rotational axis during this time period. As in the case of the real observations (Section 3.2), we assume this region to be completely open. This turns out to only slightly increase the estimated open flux (see the Appendix). Figures 8(g) and (h) show the detected CHs using these two methods (same as the black contours in (e) and (f)). Comparison with Figures 8(c) and (d) shows that the algorithms generally detect most of the true open field regions. However, 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, including 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. Overall, PSI-MIDM does a better job of identifying open flux than PSI-SYNOPTIC, especially in the polar regions (e.g., longitudes 200°–260° in the north and 150°–210° in the south), where it correctly identifies regions that are obscured from PSI-SYNOPTIC. However, PSI-MIDM overestimates the open field area relative to PSI-SYNOPTIC in other regions, such as near the southern pole at longitudes 20°–100°.

For the PSI-MIDM method, we find that the total Sun detected CH area is 83.91 × 10¹⁰ km² and the detected open flux is 285 × 10²⁰ Mx. The PSI-SYNOPTIC method finds a detected CH area of 67.98 × 10¹⁰ km² and detected open flux of 240 × 10²⁰ Mx. The true open field area was 60.41 × 10¹⁰ km² and true open flux was 386 × 10²⁰ Mx. Both methods overestimate the total areas of all CHs (38.9% by PSI-MIDM and 12.5% by PSI-SYNOPTIC) but underestimate the total open flux (PSI-MIDM captures 73.8% of the flux and PSI-SYNOPTIC captures 62.1%). These underestimates are discussed further in the following section.

Our comparison of detection methods for the area of the observed CH reveals a standard deviation of σ_A ≈ 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 σ_A ≈ 43%–46%. These uncertainties are for a particularly well-observed low-latitude CH, with no large active regions closely adjacent. There could be more variability (and therefore more uncertainty) in detections performed on more complex CHs.

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 × 10¹⁰ km²) was actually 36% greater than the true value (7.4 × 10¹⁰ 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 misidentified as open, they do not contribute as much to the flux estimate because the opposite polarities cancel in the integration.

While the standard deviation for the detected areas provides a measure of the uncertainty of the different methods, this does not necessarily imply that the mean value is the best estimate of the true value. For the detections performed on the simulated CH (Table 3), SPoCA and PSI-MIDM provide values that were smaller and further from the mean value (compared to the other three methods) for both the detected area and open flux, but these turned out to be closer to the true values for this case.

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 × 10²⁰ Mx, with a most probable value of 482 × 10²⁰ Mx, corresponding to B_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 well above the mean value 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.

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 the 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 193 and 195 Å emission; Caplan et al. 2016, see Section 4.2 and Figure 18). In Figure 10(b), even the minimal boundary area contour (blue) is larger than and almost completely envelopes the true open field contour (cyan). This occurs because 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 193 Å 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 × 10¹⁰ 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.

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(g) and (h)). 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 northern active region near ∼40° latitude and ∼260° longitude (panels (a), (b), (e), and (f) of Figure 8) inhibits the detection of the adjacent open flux regions at longitudes ∼240° and ∼280° (Figures 8(c) and (d)). These CHs account for 6% of the total open flux in the model, but they are almost completely missed in the extracted CH boundaries for both methods (panels (g) and (h) of Figure 8). Linker et al. (2017) and Caplan et al. (2019) also found that open flux was underestimated by these detection methods for the same reasons.

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 (≈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 ∼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_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 et al. 2021; Riley et al. 2021) suggests 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 et al. 2021) indicate that this process appears to be more prevalent than previously thought.