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