Effect of Additional Magnetograph Observations From Different Lagrangian Points in Sun‐Earth System on Predicted Properties of Quasi‐Steady Solar Wind at 1 AU

Modeling the space weather conditions for a near‐Earth environment depends on a proper representation of magnetic fields on the Sun. There are discussions in the community with respect to the value of observations taken at several Lagrangian points (L1–L5) in the Sun‐Earth system. Observations from a single (e.g., Earth/L1) vantage point are insufficient to characterize rapid changes in magnetic field on the far side of the Sun. Nor can they represent well the magnetic fields near the solar poles. However, if the changes in sunspot activity were moderate, how well would our predictions of the solar wind based on a single viewing point work? How much improvement could we see by adding magnetograph observations from L5, L4, and even L3? Here, we present the results of our recent modeling, which shows the level of improvement in forecasting the properties of the solar wind at Earth made possible by using additional observations from different vantage points during a period of moderate evolution of sunspot activity. As an example, we also show the improvements to the solar wind forecast from adding a single observation of the southern polar area from out‐of‐ecliptic spacecraft at −30° heliographic latitude vantage point.


Introduction
Modeling the space weather conditions for a near-Earth environment depends on an accurate representation of magnetic fields on the Sun. Observations from a single vantage point lying in the ecliptic plane (e.g., Earth) are insufficient to characterize rapid changes in magnetic field on the far side of the Sun. Nor can they represent well the polar fields because the visibility of each pole changes as Earth orbits the Sun (due to ≈7.2°orbital inclination, each solar pole is visible from Earth only for 6 out of 12 months). All current numerical models used to render the structure of the solar corona (e.g., Mikić et al., 1999) or solar wind (e.g., the WangSheeleyArge or WSA model, Arge & Pizzo, 2000) employ the Carrington rotation (CR) synoptic charts representing a distribution of magnetic fields everywhere on the solar surface. The Carrington (heliographic latitute-longitude) coordinate system on the Sun is based on the assumption of a fixed period of solar rotation at the equator (mean synodic rotation is 27.2753 days), with the first CR starting 9 November 1853 (e.g., Thompson, 2006). Such CR synoptic charts are constructed over a full solar rotation (about 27 days), by adding new observations of a visible portion of solar disk to the observations taken in early periods (e.g., Bertello et al., 2014;Ulrich & Boyden, 2006, and references therein). Thus, the part of any integral synoptic map based on observations from a single vantage point at the beginning of a CR corresponds to the observations, which are about 1 month old. A middle part of such CR map is represented by 2-3 week old observations, and the longitudes near the end of each CR map correspond to more recent data. Furthermore, depending on the time of year, one of the solar poles could be nonobservable from Earth and require a special "pole filling" (e.g., Sun et al., 2011). Thus, it should not be surprising to see the modeling predictions for the heliosphere and/or corona deviating significantly from the observed conditions, especially when the magnetic field exhibits new flux emergence and other significant evolution in areas of the Sun "hidden" from Earth view (the so-called, "far side of the Sun"). The evolution of magnetic flux in those areas of the Sun could be (at least, partially) modeled using the surface flux transport, which assumes a known differential rotation, meridional flux, and diffusion. In addition, some limited information about the new flux emergence (e.g., location and size of emergence of a large active region) could be inferred from the helioseismic observations (e.g., Liu et al., 2018). Attempts to incorporate the helioseismic ©2020. The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. information and the surface flux transport to create updated synoptic maps have shown improved model outcomes (e.g., the Air Force data assimilative photospheric flux transport [ADAPT] model, Arge et al., 2013;Henney et al., 2012). The representation of solar magnetic fields could also be improved by adding observations from different vantage points. For example, the Lagrangian points L 1 and L 5 in the Sun-Earth gravitational system had been discussed. Pevtsov et al. (2016) and Weinzierl et al. (2016) have shown that having additional observations from the L 5 vantage point would improve a short-term forecast of solar wind properties and affect the long-term trends in the derived distribution of magnetic field in the solar corona. The improvements would depend on properties (e.g., size and complexity) of magnetic flux "missed" by single-vantage observations, and in cases studied by these authors, the improvements could be about 20-30% for solar wind speed, 40-70% for electron density, and 15-25% for the magnetic field .
The comparative studies of modeling outcomes with additional data from other vantage points are complicated by the fact that currently, there are no observations, which could represent "true" distribution of magnetic fields on the Sun. Thus, Petrie et al. (2018) explored an alternative approach by employing artificial data. Here we continue exploiting this approach and investigate the effect of additional vantage points on prediction of properties of the solar wind. The goal is to investigate the degree of improvement in WSA-ENLIL model predictions, if there were additional magnetograph observations from different Lagrangian points, and to evaluate the relative importance of each Lagrangian point for improving the solar wind predictions. Although our data set can be used for evaluating the space weather effects from additional observations from any vantage point, here we concentrate on the Langrangian L * points in the ecliptic plane. Still as an example, we show the effect from a hypothetical location of recently launched Solar Orbiter spacecraft (SolO, Müller et al., 2013) when it would be below the ecliptic plane at −30°heliographic latitude vantage point.

Data and Models
We use the data set of simulated synoptic magnetograms created as described in Petrie et al. (2018). The data for these magnetograms come from a numerical model of magnetic activity developed by Schrijver (2001). The model outcome represents realistic patterns of a typical solar cycle including drift of activity belts from high latitudes at the beginning of the cycle to the equator at the end of the cycle, tilt and polarity orientation (Joy's law and Hale polarity rule). Initial flux emergence is represented by simple bipoles, with later evolution due to supergranular diffusion, differential rotation, and poleward transport by the meridional flow. Example of a magnetic time-latitude plot or "butterfly diagram" corresponding to our model data set is shown in Petrie et al. (2018), Figure 7.
For our present exercise, the parameters of the numerical Petrie et al. (2018) model of magnetic activity were selected to correspond to a "typical" solar cycle with moderate level of activity. Development of active regions in the model was relatively symmetric between the two hemispheres and across different longitudes. This data set was used to create a set of observations imitating magnetograph observations from four vantage points along the Earth orbit: Lagrangian L 1 (between Earth and Sun on Earth-Sun line), L 5 (60°, trailing the Earth on Earth's orbit), L 4 (60°leading the Earth on its orbit), and L 3 (180°, antipodal to Earth location on Earth orbit, behind the Sun on Earth-Sun line). Then, the full disk magnetograms corresponding to observations from different Lagrangian points were used to create CR (synoptic) charts corresponding to the following combination of observations: from Lagrangian points L 1 , L 1 + L 5 , L 1 + L 4 , L 1 + L 4 + L 5 , and L 1 + L 4 + L 5 + L 3 . Our procedure for merging full disk observations from different vantage points into a single synoptic map is described in detail in Petrie et al. (2018). The merging scheme uses both temporal weighting and the location of area relative to solar central meridian for each viewing point (i.e., newer observations and data from areas located closer to the central meridian have higher weight). For each CR, we also created a "master" synoptic map showing the true distribution of magnetic fields on the Sun corresponding to this rotation. The visibility of solar poles vary during the year. When observed from Earth (or Lagrangian L 1 point), the north pole is mostly invisible over a 6-month period (early December-June), and the south pole is invisible over the next 6-month period. The visibility of poles has similar patterns for observations taken in Lagrangian L 5 and L 4 , but the periods are shifted by about 2 months (e.g., for north pole to early October-April for L 5 and early February-August for L 4 ). Thus, combining the observations from three Lagrangian points (L 1 , L 4 , and L 5 ) significantly shortens the period of invisibility for each pole, as compared with a 10.1029/2020SW002448 Space Weather single vantage point observations. Adding observations from L 3 would ensure visibility of both poles all the time. In the process of creating synoptic maps, for each individual vantage point we replaced missing polar fields by the extrapolated values in the same way the pole filling is done for observations from Global Oscillations Network (GONG) (for a review of different approaches to pole filling, see Sun et al., 2011). CR synoptic maps then were used as input for the WSA-ENLIL model. For modeling of solar wind properties, we employ the version of WSA-ENLIL model (WSA V2.2 and ENLIL V2.7) currently run at the Community Coordinated Modeling Center operated at NASA Goddard Space Flight Center (GSFC). The WSA-ENLIL model is one of the standard models used by the research and forecasting communities to represent a quasi-steady flow of solar wind at different locations throughout the heliosphere. It uses a Potential Field Source Surface (PFSS) model linked with the Current Sheet Model to represent the magnetic field to about 0.1 AU and the empirical relations between the expansion factor of the magnetic field and the solar wind speed (e.g., Arge et al., 2004;Odstrcil et al., 2008;Pevtsov et al., 2016). WSA also Figure 1. Longitude-latitude plots representing different parameters for a single full solar rotation. The upper panel shows the source-surface neutral line in the solar corona as derived from WSA-ENLIL modeling, and the two lower panels show solar wind speed and the derived coronal holes. Line segments plotted in black color represent magnetic connectivity of derived coronal holes with the ecliptic plane. The input photospheric magnetic field used in WSA-ENLIL model is shown in second panel from the top. The map represents an instantaneous distribution of magnetic fields on the Sun, but for comparability with traditional synoptic maps vertical (red and white) dashed lines mark the day when a particular area will be crossing the central meridian when observed from Earth/L1. While these data are entirely model-based, the dates and the Carrington rotation numbers are selected as if the observations were taken during the period when the Sun's north pole is tilted toward the Earth. The scaling for all parameters is shown by vertical color/gray bars on the right side of each panel.
includes proximity of the field line photospheric footpoint to the nearest coronal hole boundary in it empirical solar wind speed formula.
Since there are no simultaneous measurements of magnetic field and solar wind properties from different Langrangian points, here we conduct model-to-model comparisons, when the model predictions of solar wind are compared with the predictions by the same model based on different input. Figure 1 shows the distribution of magnetic field from our model, coronal field neutral line at the source surface and solar wind properties at 1 AU. For this modeling we used a "snapshot" of solar surface with known radial magnetic fields everywhere on the Sun (including polar regions). In the following discussion, we will compare the derivations based on "observations" taken from specified viewing angles to these true distributions derived from "master" synoptic map. As a general note, the reader can see that the distribution of magnetic fields exhibits pattern typical to real solar magnetic fields: bipolar structure of "active regions" (pairs of poles of opposite polarity), the Hale polarity orientation of these bipoles with the leading polarity of each "active region" being positive/negative (white/black) in the northern/southern hemisphere, elongated shape and extended "tails" of decaying regions gradually transported to higher latitudes, and a preference for one polarity large-scale magnetic fields (negative in the north and positive in the south) at high-latitude (polar) regions. The derived speed of solar wind, and the properties of coronal holes and source-surface neutral line in the solar corona are also very similar in their appearance to true solar values.

Results
Let us now consider how the distribution of these parameters changes if we restrict the input magnetic fields to "observations" from selected vantage points and their combination. Observations from Earth/L 1 do not represent well enough the changes in synoptic map due to a development of a small bipolar active region situated in the southern hemisphere at about 270°longitude. As a result of this and other evolutionary changes in the magnetic field, which are not represented by observations from L 1 , we see a significant difference in the position of the source-surface neutral line (i.e., instead of a smooth transition of the position of the heliospheric neutral line from positive to negative latitude at about 240°longitude as shown in the upper panel in Figure 1, the synoptic map from L 1 point exhibits a jump-like change there). By adding observations from L 5 point, this discontinuity disappears. There is also a notable effect in the solar wind speed in this range of longitudes (about 249-339°in Figure 2). Figures 2 and 3 show difference (ΔV sw ) in the solar wind speed computed using synoptic maps based on observations from different L * points relative to the solar wind computed using true distribution of magnetic field ("master" synoptic map shown in Figure 1). Overall appearance of synoptic maps in these two figures indicates that on average ΔV sw decreases with the observations from the additional Lagrange points, that is, the amplitude of differences decreases when more L * observations are added, and the pixel-by-pixel gradients in the synoptic map also decrease. For observations from a single L 1 point (Figure 2a), 68% of all pixels in the synoptic map have |ΔV sw |< 23% of true V sw . When the observations from L 1 , L 5 , L 4 , and L 3 are added, 68% of all pixels in the synoptic map have |ΔV sw |<17% of true V sw (Figure 3a). Larger ΔV sw differences come from higher latitudes (Figures 2 and 3). Limiting the latitudinal range to ±15°yields the following 1 σ estimates for |ΔV sw |: < 16% (L 1 ), 14-15% (L 1 + L 5 + L 4 ), and 7% (L 1 + L 5 + L 4 + L 3 ). Most of these differences are larger that the uncertainties in the modeled solar wind speed. Pevtsov et al. (2015) estimated the uncertainties for several parameters modeled using WSA-ENLIL based on the uncertainties of input synoptic maps for two solar rotations with different level of solar activity. The (1 σ) uncertainties in the solar wind speed were found to be approximately 20-30 km s −1 , or within 4-7% of an amplitude of the solar wind speed.

Space Weather
The differences between "true" and L point specific solar wind may vary depending on the magnetic connectivity. Thus, for example, the synoptic map of derived coronal holes (Figure 1, lower panel) shows that a wide area in the ecliptic plane between approximately 290°and 30°of longitude (left side of the synoptic map) is magnetically connected to a large coronal hole in the northern hemisphere. The simulations based on the "magnetograms" from all Lagrangian points show a similar connectivity, albeit the data from (L 1 + L 5/4 and L 1 + L 5 + L 4 + L 3 exhibit a better match with the "ground truth." The ΔV sw difference in the solar wind speed in the ecliptic plane for this range of longitudes is quite small, and it does not exceed a few percent (Figure 3c). For L 1 only observations, the difference increases to about 20% near the edges of this longitudinal range, which is in agreement with the difference in the magnetic connectivity derived from L 1 only data. This is also evident from Figures 2a-2c and 3a, as patterns in ΔV sw , which are present in the range of ±30°latitude and 280°-20°longitude (left side of the synoptic maps) become more uniform with progression from L 1 (Figure 2a) to L 1 + L 5 + L 4 (Figure 2c), and L 1 + L 5 + L 4 + L 3 (Figure 3a). Adding L 5 observations has the largest effect; the improvements from adding L 4 data are more incremental.
In the longitudinal range of approximately 150°and 250°, the ecliptic is magnetically connected to several small coronal holes situated in the southern hemisphere. Due to the orientation of solar poles during this period of time (north pole is visible from Earth/L 1 , and southern pole is largely invisible), the magnetic connectivity with the southern hemisphere is not well represented by the L 1 , L 5 , and L 4 data. The simulated magnetic connectivity based on data from these Lagrangian points does not reproduce well "true" connectivity. A simulated solar wind for this longitudinal range deviates significantly (up to 40% difference) from its "true" value ( Figure 3c).
The top views of the heliosphere (looking down from the direction of solar north pole) corresponding to the same simulations are shown in Figures 4-6. When using the full Sun "master" synoptic map (Figure 4), there are three distinct solar wind streamers: two on the opposite side of the Sun with respect to Earth's location (light yellow and blue colors), and one near the Earth's location (dark green color, to the left from Earth and white curved line). For the discussion purpose, we will refer to these streamers as S1, S2, and S3. S1 (yellow) has radial velocity (V r ) of about 550 km/s. For S2 (blue), it is about 280 km/s, and for S3 (green), the average velocity is about 450 km/s. When using only observations from L 1 (Figure 5, upper panel), the S1 streamer appears stronger (V r ≈ 600-650 km/s). S2 is largely unchanged, and S3 weakens significantly (with average velocities of about 250 km/s). Adding observations from L 5 ( Figure 5, low panel) weakens S1 (about 400 km/s), but slightly enhances S3 (V r ≈ 500 km/s). With the additional observations from L 4 (not shown), S1 becomes a little bit stronger, but S2 and S3 are weakened slightly. In the model outcome based on observations from four Lagrangian points (L 1 + L 5 + L 4 + L 3 , Figure 6, upper panel), the patterns of S1, S2, and S3 streamers get closer in amplitude of solar wind to the prediction based on the full Sun "master" synoptic map. In all cases, the polarity of the IMF in the vicinity of Earth is negative. However, the prediction based on L 1 observations only, significantly shifts the location of the current sheet away from Earth both in longitude in the ecliptic plane and in latitude in the Sun-Earth plane perpendicular to the ecliptic (compare Figure 4 and upper panel in Figure 5). Moreover, the prediction based on L 1 observations significantly underestimates the solar wind speed at high latitudes (in this case, in the southern hemisphere). We think this is related to a much weaker high-latitude/polar fields in the southern hemisphere, and a poor visibility of pole from the L 1 location during the time of the CR selected for this study.
While the observations from L * points do improve the visibility of solar poles, one can argue that the observations from a spacecraft out of ecliptic plane would be a much better solution. As one example, recently Figure 3. Difference in solar wind speed between the "ground truth" and the synoptic magnetic field maps from (a) L 1 + L 5 + L 4 + L 3 and (b) L 1 + SolO vantage point corresponding to its heliographic location at −30°l atitude and 99°longitude. Panel (c) shows the difference between the "ground truth" solar wind speed ΔV sw along the solar equator and the WSA-ENLIL reconstructions based on other vantage points: L 1 (black), L 1 + L 5 (red), L 1 + L 5 + L 4 (blue), L 1 + L 5 + L 4 + L 3 (green), and L 1 + SolO (brown).

Space Weather
launched Solar Orbiter spacecraft (SolO, Müller et al., 2013) will eventually take the observations of the Sun at about ±30°of the heliographic latitude above and below the ecliptic plane. While the analysis of out-ofecliptic orbits is possible with our model of magnetic Sun, this would require a separate paper because their orbits are so different from those that are analyzed here, with fast changes in radius and the latitude as well as the longitude. The contribution of these observations to synoptic maps would be complex and nonuniform because of the fast-changing perspectives. As a demonstration, Figure 6 provides an example of solar wind modeled using combined observations from Lagrange L 1 and SolO vantage corresponding to its heliographic location at −30°latitude and 99°longitude (middle of the CR). For this test, we assumed 1 AU distance between SolO and Sun. While SolO vantage provides a better view of the southern pole, which is not visible from the L 1 location for the time corresponding to the modeled CR, the longitude of SolO spacraft is close to the longitude of L 1 . Also, for this modeling exercise, we are using a single observation from SolO's vantage point. Perhaps as the result of this limitation, the L 1 + SolO synoptic map is quite similar to L 1 map, and thus, we see only a very small difference in solar wind properties based on the L 1 and L 1 + SolO observations (compare Figures 2 and 3b and for the top view, Figures 4-6, low panel). When adding out-of-ecliptic observations from SolO, the improved visibility of the southern pole reduces ΔV SW difference in high latitudes in the southern hemisphere. There the differences become more similar to L 1 + L 5 + L 4 simulations (compare Figures 2c and 3b). There are also small improvements in low latitudes of the solar wind speed (ΔV SW ) as compared with L 1 data (Figure 2a). We speculate that the improvements in low latitudes could be larger if one selects a larger longitudinal separation between Lagrangian L 1 and SolO locations, but here we use this only as a demonstration that the approach we describe in this paper could be applied to SolO or any other out-of-ecliptic plane observations. A detailed study of improvements from a combination of L 1 and SolO observations on WSA-ENLIL modeling will be done in a separate future paper.
The improvements to the solar wind parameters as modeled by WSA-ENLIL modeling with the additional observations from various Lagrangian points described in this article represent the best case scenario. We only consider the effect of different viewing angles and ignore the inequities in the noise level across the solar disk. Thus, for example, the noise is higher near the solar limb. This difference in noise properties is not currently taken into consideration, when merging the observations from different Lagrangian points to create a single synoptic map. Similarly, while the observations taken from different Lagrangian points do improve the visibility of polar regions, the near-the-pole observations are still the subject of higher noise level, and thus could be considered less trustful. We note, however, that even with this known issue of higher noise level near the poles, the current observations from L 1 still allow deriving a useful information about polar magnetic fields (e.g., GONG, HMI synoptic maps). As a mitigation, the future instruments could also implement additional approaches such as longer time exposure, larger pixel size, and use of full vector field observations to improve the inference about the polar magnetic fields. We plan on addressing these and other shortcomings of our current modeling in the follow-up future papers.

Space Weather
This article describes the effects of observations from Lagrangian L * points only. These points provide a more stable location for a long-term observations, which arguably is the benefit for a space weather operational forecast. Still, our results are applicable to spacecraft on a STEREO-like orbit, which location gradually changes (drifts) relative to Earth. The STEREO satellites separate from Earth at a rate of about 22°per year, or 1.6°per rotation. Thus, the STEREO-like drifter scenario can be inferred to some degree from the various combinations of Lagrange point results. The observations from the STEREO-like drifters may have both advantages (e.g., in situ measurements in the ecliptic plane which may be more easy to compare with near-Earth/L 1 observations) and disadvantages (e.g., no improvements in polar coverage for the small longitudinal separation). Those need to be evaluated separately, perhaps as part of a new mission requirements, which is outside of scope of our present article.

Conclusions
Even for a moderate level of solar activity (e.g., no significant flux emergence on the far side of the Sun), adding magnetograph data from different viewing angles results in significant improvements in solar wind speed prediction. The improvements for locations in the ecliptic plane are likely to be the result of better longitudinal coverage, while the improvements in high latitudes (outside ecliptic plane) are likely due to better viewing angles for solar polar areas. L 1 + L 5 + L 4 + L 3 observations yield the best agreement with the full Sun observations. In respect to solar wind predictions at L 1 /Earth, adding L 5 data yields the largest improvement. The improvements from adding L 4 data are smaller. This could be related to the fact that the time evolution of magnetic flux on solar surface (excluding new flux emergence) is longer than the time it takes for the Sun to rotate between L 1 and L 4 viewing angles. As a result, the L 4 update to a synoptic map based on L 1 + L 5 observations is relatively minor. In comparison, the L 5 update to the map based on L 1 observations is much more significant. The overall degree of "freshness" of the map is the same in the L 1 + L 4 and L 1 + L 5 cases, but the L 1 + L 5 keeps the Earthward side fresher than the L 1 + L 4 case does, which gives a better prediction at Earth. With L 1 + L 4 observations, the oldest data, east of the L 1 field of view, affect the earthward side and prediction at Earth, whereas with L 1 + L 5 the oldest data are beyond the L 5 field of view, eastward, well out of sight of Earth, and having much less impact on the prediction at Earth.
Our results demonstrate that the degree of improvement may depend on a distribution of magnetic field on the Sun. For some areas, the improvement could be significant, while in others it could be only marginal. The combined observations from four Lagrange points (L 1 , L 5 , L 4 , and L 3 ) provide the most consistent improvement in respect to the "ground truth." If one is limited to a single additional viewing point, L 5 seems to provide the most significant improvement to L 1 only observations. Improvements from additional observations from L 4 are more incremental. We note, however, that these conclusions only concern the solar wind predictions. The observations from L 4 could be more important than the L 5 observations for studying the effects of the Solar Energetic Particles on Earth orbiting satellites.
The results of WSA-ENLIL modeling reported here are based on a single rotation during a moderate level of solar activity. Significance of improvements under a broader range of surface conditions requires further investigations.