in-situ measurements from well-separated heliospheric spacecraft inside 2.5 AU give self-consistent results of total open heliospheric flux. However, they found that open flux estimates from multiple spacecraft at the same R increasingly vary as R increases, with the largest discrepancies occurring beyond 2.5 AU (Owens et al. 2008a, Figure 1). They suggested that this could be attributed to a reduced signal-to-noise ratio in spacecraft detection of \( B _ { \mathrm { r } } \) further away from the Sun due to tangential and meridional fluctuations along each magnetic field line that result in these field components bleeding into \( B _ { r } \) measurements (Owens et al., 2008a). In a later set of articles, Lockwood and Owens (2009) and Lockwood, Owens, and Rouillard (2009a,b) revisited this topic and attribute what they term the “flux excess effect” not to small-scale structure introduced by the field line’s own propagation, but to separate large-scale longitudinal solar wind structures that distort the magnetic field (Lockwood, Owens, and Rouillard, 2009b, Figure 1). They derived a correction term to apply to spacecraft \( B _ { r } \) measurements to account for these kinematic effects, and when this is applied, for the time period investigated in their study they produce good agreement between PFSS and spacecraft-derived open flux (Lockwood, Owens, and Rouillard, 2009b, Figure 9). Owens et al. (2017) accounted for this same kinematic contribution to \( B _ { \mathrm { r } } \) instead using observations of sunward suprathermal electron beam or “strahl” to filter out instances where field lines were locally inverted at 1 AU. Their method resulted in higher open fluxes than the kinematic correction, closer to results obtained when averaging \( B _ { \mathrm { r } } \) over one day before taking the unsigned value. Both methods on average fell within the results of averaging \( B _ { \mathrm { r } } \) between one to five days before taking the unsigned value (Owens et al., 2017, Figure 5); however, unlike the temporal averaging technique, these methods attempt to account for the flux excess with more physical justification.

Another way that the differences between both model-derived and spacecraft-observed open flux have been explained is by applying correction factors to the photospheric-field maps to drive coronal models. Wang, Lean, and Sheeley (2000) compared estimates of total open flux from 1971 – 1998 using both the PFSS model and spacecraft observations and found, with a few notable exceptions (e.g. 1978 – 1980 and 1985 – 1989), that they agree remarkably well when using a latitude [λ] dependent saturation correction factor. The photospheric field used to derive the open flux was a combination of traditional Carrington photospheric magnetic- field synoptic maps from Mount Wilson (MWO) and Wilcox (WSO) Solar Observatories. It was necessary to use WSO maps from 1976 – 1995 because the MWO magnetograph suffered from instrumentation issues during this period. They also assumed, based on previous work (Wang and Sheeley, 1995), that the MWO correction factor \( ( 4 . 5 \! - \! 2 . 5 \mathrm { s i n } ^ { 2 } \, \lambda \)) was equally applicable to both the MWO and WSO observations, as both the instruments are very similar to each other and use the same Fe I 5250 A˚ line to derive the magnetic field. However, Svalgaard (2006) challenged the application of the MWO saturation correction to WSO data, arguing that it is specific to the MWO instrument and that a constant correction factor of 1.85 (updated from Svalgaard, Duvall, and Scherrer 1978, originally 1.8) is the appropriate correction factor for WSO data. Other studies, spanning several years (Owens et al., 2008b, Linker et al., 2016) and focusing on one Carrington rotation (Linker et al., 2017), used photospheric-field maps

derived from other observatories and found similar discrepancies between model-derived and spacecraft-measured \|Br\|. In all of these studies, in-situ observations on average produced larger \|Br\| estimates than models, except Linker et al. (2016) where a correction factor of 1.5 was applied to the photospheric-field maps for the results to agree.

Riley (2007) separately applied both a constant 1.8 scaling factor and the MWO latitude-dependent correction to WSO input maps from 1976 – 2007 and calculated total open flux using the PFSS model for both sets of maps. He found that the maps with the MWO saturation correction factor applied to them pro-duced the best overall agreement with the heliospheric derived fluxes but noted significant mismatches during the periods of solar minimum, exactly when one would expect the agreement to be best. For the case of the 1.8 saturation factor applied to the WSO maps, the open-flux results agreed well with the heliospheric values only during solar minimum and under-predicted them for all other times. To explain this Riley proposed and tested the idea that interplanetary coro-nal mass ejections (ICMEs) propagating out into space yet still magnetically connected back to the Sun are detected by spacecraft in the same way that coronal hole open flux is. Riley argued that the total unsigned heliospheric flux is comprised of open coronal hole and closed ICME fluxes, especially during solar maximum. This assumption is reasonable, since distinguishing between open field lines and closed ones from ICMEs still rooted back at the Sun in spacecraft observations, especially when the front of the ICME is far beyond 1 AU, is generally very difficult. Using a simple model to estimate heliospheric ICME flux from sunspot number, Riley found an extremely good match between the combined ICME plus WSO open-flux estimates and heliospheric values. While not claiming to have settled the issue, Riley’s results nonetheless provided an alternative explanation for the temporal variation of observed heliospheric flux without having to resort to correcting the WSO magnetic field data with the MWO saturation correction factor. Owens et al., in a series of articles (Owens and Crooker, 2006; Owens et al., 2007; Owens, Crooker, and Lockwood, 2011), similarly argued that much, if not all, of the variation of heliospheric flux can be explained by a constant open solar magnetic flux baseline or floor with the addition of a time-varying, solar-cycle-dependent closed ICME magnetic-flux component.

Alternatively, total open flux can be determined by identifying coronal holes in chromospheric or coronal observations. Coronal hole areas are calculated from observations after their boundaries are determined either through manual (Harvey and Recely, 2002) or automated (e.g. Henney and Harvey, 2005; Scholl and Habbal, 2008; Krista and Gallagher, 2009; Lowder et al., 2014; Verbeeck et al., 2014; Boucheron, Valluri, and McAteer, 2016; Caplan, Downs, and Linker, 2016; Garton, Gallagher, and Murray, 2018; Hamada et al., 2018) methods. To derive open flux, coronal-hole contours are paired with corresponding synoptic photospheric-field maps. The advent of high- resolution coronal EUV images allows for an independent estimate of the open flux in comparison to model estimates and in-situ observations for the past decade. Observationally derived coronal holes also provide constraints on model-derived coronal hole boundaries.

In this article, we estimate total open magnetic flux from 1990 – 2013 in two different ways using global photospheric magnetic-field maps derived from