Uncertainty Estimates of Solar Wind Prediction using HMI Photospheric Vector and Spatial Standard Deviation Synoptic Maps

sec:intro

The solar magnetic field plays a significant role in controlling the solar wind outflow, and it is well–known that the solar wind behavior is greatly influenced by the shape of individual bundles of magnetic field lines (Zirker, 1977a, b; Levine, Altschuler, and Harvey, 1977; Wang and Sheeley, 1990; Fisk, Zurbuchen, and Schwadron, 1999a, b; Kojima et al., 2007; Cranmer, van Ballegooijen, and Woolsey, 2013; Krieger, Timothy, and Roelof, 1973; Riley, Linker, and Arge, 2015). However, the mechanisms that give rise to the observed slow and fast solar wind streams – the two distinct components with differing physical properties – are still not understood satisfactorily. This is mainly due to lack of direct measurements of near–Sun solar wind properties, except for the Parker Solar Probe measurements released in November 2019, and limited observation of coronal magnetic field (e. g. Bak-Steslicka et al., 2013; Rachmeler et al., 2013; Dove et al., 2011). Much of our current understanding is based on, in addition to the photospheric observations of the Sun and near–Earth solar wind measurements, the magnetohydrodynamic (MHD) models and the magnetostatic models of the corona such as the Potential Field Source Surface (PFSS: Altschuler and Newkirk, 1969; Schatten, Wilcox, and Ness, 1969) and the Current Sheet Source Surface (CSSS: Figure \ireffig:geometry; Zhao and Hoeksema, 1995) models.

The current solar wind prediction scheme is built on the inverse correlation (the well-known Wang & Sheeley empirical relation) between the rate of expansion of the magnetic flux tubes (FTEs: described in detail below) in the inner corona (below 2.5 $R_{\odot}$), computed using coronal extrapolation models, such as CSSS and PFSS models, and the observed solar wind at 1 AU. These models extrapolate the observed photospheric magnetic fields to deduce the coronal and the heliospheric magnetic field (HMF) configuration, making use of the photospheric flux density synoptic maps (top panel in Figure \ireffig:hmi_maps) constructed from the magnetograms measured by ground-based and spacecraft observatories as the lower boundary conditions. These synoptic maps are among the most widely-used of all solar magnetic data products. Moreover, as well-known, the quality and reliability of these synoptic maps are critical to the accuracy of solar wind predictions (speed & magnetic field), determination of locations of coronal holes (CH) and the magnetic neutral line (NL), and the global structure and properties of many other solar and heliospheric phenomena (e. g. Riley et al., 2014; Arge and Pizzo, 2000). Therefore, the uncertainties in the model predictions that are caused by the uncertainties in the synoptic maps are worthy of study. An estimate of the uncertainties in the construction of these synoptic maps were not available until recently when Bertello et al. (2014) obtained the spatial standard deviation synoptic maps (bottom panel in Figure \ireffig:hmi_maps). For each photospheric synoptic maps, there are 98 Monte-Carlo realizations of the spatial standard deviation maps. Instrumental noise in the magnetic field measurements, the spatial variance arising from the magnetic flux distribution and the temporal evolution of magnetic field contribute to these uncertainties. Bertello et al. (2014) have shown that these uncertainties led to significant differences in the CH and NL locations obtained using the PFSS model.

In the theory of solar wind by Parker (1958), there exists a reference height above the photosphere beyond which the magnetic field is dominated by thermal pressure and inertial force of the expanding solar wind. To mimic the effects of the solar wind expansion on the field, an equipotential upper boundary above the photosphere (a source surface with radius $R_{ss}$) has been introduced in the PFSS model, forcing the field to be open and radial on this surface (Altschuler and Newkirk, 1969; Schatten, Wilcox, and Ness, 1969; Schatten, 1971). Following Hoeksema (1984), it is customary to place the source surface $R_{ss}$ at 2.5 $R_{\odot}$ although other choices have led to more successful reconstructions of coronal structures during different phases of the solar cycle (Lee et al., 2011; Arden, Norton, and Sun, 2014). The lower boundary has been identified with the photosphere. Assuming the region between these two boundaries to be current–free, the PFSS model solution is uniquely determined in the domain $R_{\odot}$$\;\leq r\leq R_{ss}$. The PFSS model is the simplest and the most widely–used global coronal model. The open–field footpoints obtained by PFSS model can be compared with observed coronal holes while the neutral lines at the outer boundary can be compared with observed streamer locations. It serves as a useful reference for more sophisticated models.

Though still debated, the reference height where the plasma takes over the magnetic force, is considered to be $\sim 10-20$ $R_{\odot}$ (Schatten, 1971; Zhao and Hoeksema, 2010). Moreover, though magnetic field lines are open above a height around 2.5 $R_{\odot}$, the coronal field is not radial, as evident from many observations, until farther out in the corona. In the CSSS model, these two heights are represented by a cusp surface and a source surface as shown in Figure \ireffig:geometry, typically placed at 2.5 and 15 $R_{\odot}$, respectively. Moreover, the field lines are allowed to be non–radial between the cusp surface and the source surface though they are all open at the cusp surface (Zhao and Hoeksema, 1995). This is consistent with observations of the Large Angle and Spectrometric Coronagraph (LASCO) instrument on board the Solar and Heliospheric Observatory (SOHO), which revealed that field lines, except near the magnetic neutral line, are non–radial until further out in the corona (e. g., Zhao, Hoeksema, and Rich, 2002; Wang, 1996). Further, the corona is not strictly current–free as evidenced by the numerous structures and features of the corona (Zhao and Hoeksema, 1995; Hundhausen, 1972; Pneuman and Kopp, 1971). The volume and sheet currents in the lower corona couple with the magnetohydrodynamic forces to produce the distension of coronal magnetic field lines into an open configuration – the solar wind, the helmet streamers and the coronal holes are all evidence of the existence of these currents in the corona. Coronal currents, being complex and widely distributed, are modeled as volume currents while the heliospheric current sheet is modeled as a sheet current (Zhao and Hoeksema, 1995; Hundhausen, 1972; Pneuman and Kopp, 1971), maintaining the total pressure balance between regions of high and low plasma density. The CSSS model incorporates volume and sheet currents, based on the analytical solutions obtained by Bogdan and Low (1986) for a corona in static equilibrium, assuming that the electric currents are flowing horizontally everywhere. The lower boundary is taken as the photosphere as in the PFSS model.

Owing to the treatment of electric currents in the model and the particular geometry, the CSSS model is considered to depict a more realistic scenario of the solar corona than the PFSS model. Particularly, based on the fact that the magnetic field lines are allowed to be non–radial between the cusp surface and the source surface, the CSSS model is expected to map coronal features or the coronal sources of fast/slow wind back to the photosphere with greater accuracy than the PFSS model. The CSSS model has been shown to predict $\rm{V}_{\rm{sw}}$ and HMF with greater accuracy than the PFSS model (Zhao and Hoeksema, 1995; Poduval and Zhao, 2014; Poduval, 2016). Also, Zhao, Hoeksema, and Rich (2002) used it to map the radial evolution of helmet streamers and found that their results were in general agreement with observations. Moreover, Schuessler and Baumann (2006) have shown that the Sun’s open magnetic flux computed using CSSS model is more accurate than that computed using PFSS model. Further, Dunn et al. (2005) and Jackson et al. (2015) found that the coronal field and the north–south components of HMF computed using the CSSS model matched reasonably well with spacecraft observations.

The causes of the discrepancies between the current predictions of space weather events such as the arrival times of CMEs and high speed streams (HSS) at Earth, and the actual observed times (e.g. Pizzo et al., 2011) remain to be investigated in detail using novel approaches for better accuracy of these predictions. One critical but unavailable information in these predictions is an estimate of uncertainties. In this paper, we present the results of an investigation of how the uncertainties in the construction of photospheric synoptic maps affect solar wind prediction. For this, we obtained an estimate of the uncertainties in the prediction of $\rm{V}_{\rm{sw}}$ as the root mean square error (RMSE) between the predicted and the in situ observations (from the OMNI database) of solar wind speed. Further, we compared the predicted CH and NL locations with those derived from the EUV data the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI) telescopes on board the Solar TErrestrial RElations Observatory (STEREO) to present the extent of variations in the computed CH and NL locations in relation to the observations. As a first step, we selected three Carrington rotations representing the minimum (2010, 3–30 October: CR2102), the late-ascending (2013, 14 May – 11 June: CR 2137) and the early-descending (2015, 1–28 February: CR 2160) phases of the solar cycle, in an attempt to explore and present the significance of uncertainty estimate in the solar wind prediction. We expect the present exploratory work will lay the foundation for future comprehensive analyses.

In § \irefsec:data, we present a description of the HMI vector synoptic maps and the derived standard deviation maps, and all other data used for the analysis in this paper. The method we adopted and the results are described in § \irefsec:method and a discussion of the results is presented in § \irefsec:results, followed by our concluding remarks in § \irefsec:discussion.

sec:data

The HMI synoptic maps are available from CR 2096 (since 6 May, 2010) and the spatial standard deviation synoptic maps from CR 2006 (29 January – 24 February, 2005) till present. We used the regular vector/longitudinal synoptic maps (processed by the NSO pipeline: Synoptic Maps based on SDO/HMI observations, https://solis.nso.edu/0/vsm/vsm_maps.php) and the corresponding ensemble of flux density synoptic maps consisting of 98 Monte-Carlo realizations of the spatial standard deviation maps (described in § \irefsub:variance and depicted in Figure \ireffig:hmi_maps) for three Carrington rotations. For simplicity, we refer to these 98 synoptic maps as “MCRs” and the regular magnetic flux density synoptic maps as “HMI synoptic maps” for the remainder of this paper.

We selected CR 2102 (3 – 30 October 2010) which falls within the extended minimum phase, CR 2137 (14 May – 11 June, 2013), during the late-ascending phase and CR 2160 (1 – 28 February, 2015) during early-descending phase of the solar cycle 24. During rotations CR 2102 and 2160, there were no Interplanetary Coronal Mass Ejections (ICMEs: Cane and Richardson, 2003; Richardson and Cane, 2010) reported http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm; Courtesy Dr. Ian Richardson) while in CR 2137, there were a few solar flares and CMEs observed. The PFSS and CSSS models are static models and therefore, do not incorporate the effects of transients such as solar flares and CMEs.

The CSSS computation for a single Carrington rotation takes about 8 hours on a laptop/desktop. For each Carrington rotation selected for the present study, there are 98 MCRs (equivalent to 98 Carrington rotations) of the spatial standard deviation maps. Therefore, the solar wind prediction using all the MCRs for the three Carrington rotations selected takes many days of CPU time, achieved in a reasonable time-span with the help of a start-up allocation on Comet at the San Diego Supercomputer Center (SDSC) of the National Science Foundation’s Extreme Science and Engineering Discovery Environment (XSEDE: Towns et al., 2014).

To determine the uncertainties in the model predictions, we compared the simulated CH and NL locations with the observed coronal holes and streamer locations (Jin, Harvey, and Pietarila, 2013; Petrie and Patrikeeva, 2009), and compared the predicted $\rm{V}_{\rm{sw}}$ with in situ measurements near 1 AU. For this, we used the coronal synoptic maps derived from the full–disk EUV images in 195 Å data from STEREO/SECCHI telescopes and the multispacecraft compilation of solar wind data from the OMNIweb archives (http://omniweb.gsfc.nasa.gov). All these data are publicly available.

sec:method

The uncertainty estimate of solar wind prediction presented in this paper is based primarily on the computations of the CSSS model. We obtained the global coronal magnetic field by extrapolating the observed photospheric magnetic field (the ensemble of HMI magnetic flux density synoptic maps) to the corona (and beyond) using the CSSS model and subsequently, the FTEs according to Equation \irefeq:fte and predicted $\rm{V}_{\rm{sw}}$ at 2.5 $R_{\odot}$ using the empirical relationship shown in Table \ireftab:speedfte. We then propagated the predicted $\rm{V}_{\rm{sw}}$ kinematically to 1 AU to compare with the OMNI data of observed solar wind..

We computed the footpoint locations of the open field lines (CHs) and the magnetic neutral lines (NLs) (Figures \ireffig:chnl1 – \ireffig:chnl3) using the CSSS model and compared with those computed using the widely used PFSS model. We utilized the numerical elliptic solver available in MUDPACK of National Center for Atmospheric Research (http://www2.cisl.ucar.edu/resources/legacy/mudpack)) finite–difference package: (see Petrie, 2013, for details) for the PFSS computations. Such a comparison will provide additional validation of the predictive capability of the CSSS model (see Poduval and Zhao, 2014; Poduval, 2016, for CSSS model validation.). In all the model computations, we used the “HMI synoptic maps” and the corresponding “MCRs” as the lower boundary conditions for each of the three Carrington rotations selected for the study.

While a complete 3-dimensional MHD model may represent the corona more realistically, magnetostatic models such as the PFSS and the CSSS models are computationally inexpensive and much faster. Radial lower boundary data from photospheric vector measurements are appropriate for this problem because they give us genuine observations of the radial flux distribution, which is not the case with the longitudinal field measurements usually employed in global coronal/heliospheric modeling.

For the solar wind prediction, we adopted a two–step method similar to Poduval and Zhao (2014) and Poduval (2016) with the exception that instead of mapping the observed solar wind back to the corona, we computed the $\rm{V}_{\rm{sw}}$ at 2.5 $R_{\odot}$ and propagated kinematically to 1 AU for comparison with in situ observations.

In Step 1, we computed FTEs on a Carrington longitude–heliographic latitude grid of $1^{\circ}$ at 2.5 $R_{\odot}$ corresponding to the cusp surface in the CSSS model and “predicted” $\rm{V}_{\rm{sw}}$ by making use of the WS empirical relationship shown in Table \ireftab:speedfte (Wang and Sheeley, 1990; Wang, 1995; Wang et al., 1997). We computed FTE at the cusp surface (at 2.5 $R_{\odot}$) because the magnetic flux tube expansion rate that influences the solar wind speed is relevant and significant at the base of the corona where the magnetic field lines begin to open, namely, the cusp surface (Poduval, 2016). The field lines become radial at the source surface at 15 $R_{\odot}$. Then we obtained a quadratic function (Figure 2 in Poduval and Zhao, 2014) that best fitted the pair, predicted–speed/computed–FTE. We adopted this method because the quadratic equation best represents the nonlinear, super–radial expansion of the magnetic flux tubes and is physically intuitive, though simple. This approach is different from that of Arge and Pizzo (2000), modified in McGregor et al. (2011). Moreover, as shown in (Poduval, 2016), the temporal variation of the coefficients provides a strong implication of the influence of the changing magnetic field conditions on the solar wind outflow.

In order to obtain the coefficients of the best–fit quadratic equation and the subsequent solar wind speed prediction for the three selected Carrington rotations, CRs 2102, 2137 and 2160, we used the original NSO processed HMI synoptic maps over a four–Carrington rotation period that included the respective CRs chosen for the present study. Our Step 2 consisted of predicting solar wind speeds using the FTEs computed for each of 98 Monte-Carlo realizations of the standard deviation maps described earlier and the fitted quadratic equations to obtain the predicted $\rm{V}_{\rm{sw}}$ at 2.5$R_{\odot}$ for each of the selected rotations.

For the forward propagation of predicted near–Sun solar wind, we adopted a simple, kinematic model described in Arge and Pizzo (2000) allowing for interaction between neighboring fast and slow streams to a limited extent. Using this approach, a solar wind velocity synoptic map, V–map (Figure \ireffig:vmap), is created at the source surface with the predicted $\rm{V}_{\rm{sw}}$ as described in § \irefsec:method. Then the solar wind is allowed to propagate at constant radial speed, the value computed at each grid point at the source surface, for a distance of 1/8 AU. At this point, the velocities were recalculated to allow for the interaction between fast and slow winds according to:

where, $v_{i}$ is the solar wind speed at the $i^{th}$ grid. The new velocities are now used for propagating the solar wind to 2/8 AU, where the velocities are recalculated according to Equation \irefeq:interact. This is continued until the solar wind reaches 1 AU.

For obtaining the prediction accuracies, we used the daily averaged solar wind speed from the OMNI archive corresponding to the arrival times of the predicted solar wind. Since our model predictions are at a higher resolution, we took a $13^{\circ}$ average near the ecliptic corresponding to the observed $b0$ angle available in the OMNI data for better comparison. Figure \ireffig:swspred shows the uncertainties (the spread) in the predicted solar wind speed for the 98 MCR of the HMI spatial standard deviation maps for the three selected Carrington rotations. We, then, obtained the RMS errors between the CSSS predictions and the observed solar wind (Figure \ireffig:swserr) for all the three Carrington rotations.

sec:results

Presented in this paper is the influence of the uncertainties in the construction of photospheric magnetic flux density synoptic maps from the observed daily magnetograms on the coronal features and solar wind predicted using them. These uncertainties are represented as the spatial standard deviation maps as shown in Figure \ireffig:hmi_maps. As well known, the flux density synoptic maps serve as the inner boundary conditions for various coronal models (e.g. Hoeksema, 1984; Riley, Linker, and Arge, 2015; Linker et al., 2017), including the operational WSA model (Arge and Pizzo, 2000), for computing the coronal and interplanetary magnetic fields, coronal features and the solar wind speed. In this study, we obtained the predicted CH locations, NLs and $\rm{V}_{\rm{sw}}$ at 1 AU using the coronal extrapolation models, CSSS & PFSS, and the spatial standard deviation synoptic maps (Bertello et al., 2014) derived from the 12–minute averaged full–disk SDO/HMI longitudinal magnetograms (m_720 series (Scherrer et al., 2012)) and the fully disambiguated vector magnetograms (b_720 series (Hoeksema et al., 2014)) through the NSO SOLIS/VSM pipeline.

Figures \ireffig:chnl1 – \ireffig:chnl3 depict the locations of CHs and NLs computed using the ensemble of HMI spatial standard deviation maps for CR 2102, 2137 and 2160. The left columns show the results of PFSS model while the right columns represent the CSSS model. A visual inspection of the SECCHI 195 Å synoptic map reveals that the locations of CHs match reasonably well with the predicted CH locations. Also, the CH locations and the NLs predicted by the CSSS model matches with the predictions of PFSS model in general. However, we note that the northern high–latitude CH around ($90^{\circ},60^{\circ}$) is larger in the PFSS than the CSSS model, whereas the low–latitude CH structure around ($80^{\circ},0^{\circ}$) is more extensive in the CSSS than the PFSS model. These general agreement between the models and, between the model and observations validates the robustness of the CSSS model in predicting the coronal features and the $\rm{V}_{\rm{sw}}$. Moreover, we note that there is a large spread in the computed neutral lines and a few are too far outside of the general trend of the neutral lines. Interestingly, the spread in the NLs is minimum for CR 2160 which is close to the maximum phase of the solar cycle, contrary to the general expectations. The NLs and CHs are key features of the coronal models and the spread in the NL and the CH locations as seen in Figures \ireffig:chnl1 – \ireffig:chnl3 indicate the influence of the uncertainties in the photospheric magnetic field measurements on these features.

The top panel of Figure \ireffig:vmap depicts the synoptic map of the predicted solar wind speed, V-map, at 2.5 $R_{\odot}$ for CR 2160 (1 – 28 February, 2015). Here, the slow solar wind (speed $\leq 450$ km $\rm s^{-1}$) is depicted by red and the fast wind (speed $>750$ km $\rm s^{-1}$) is represented by dark blue. The bottom panel shows the coronal magnetic fields modeled by the PFSS model using the Wilcox Solar Observatory synoptic maps (Courtesy: J. T. Hoeksema, http://wso.stanford.edu/synsourcel.html. The black solid line represents the heliospheric current sheet (HCS). The slow wind belt of the predicted solar wind speed generally follows the HCS as evidenced by a visual inspection of the two figures. Since the WSO synoptic maps have been extensively used as a standard for validating model predictions of global coronal structures for decades, this comparison provides further validation of the CSSS model in the prediction of solar wind.

Figure \ireffig:swspred depicts a comparison of $\rm{V}_{\rm{sw}}$ predicted by CSSS model (gray solid lines) for each of the 98 spatial standard deviation maps for all the three Carrington rotations, CR 2102 (3 – 30 October, 2010), 2137 (May 14 – June 11, 2013) and 2160 (1 – 28 February, 2015), with the corresponding observed (OMNI) solar wind speed (black solid curve). The green line depicts the CSSS prediction for the regular HMI synoptic maps. The blue and orange circles represent the maximum and minimum envelopes in the ensemble prediction. The predictions are spread over a large range and though the prediction for the regular synoptic map has a large deviation, several of the standard deviation maps seem to have predicted the solar wind much closer. A possible reason for “missing” the two high speed streams in CR 2137 could be the difficulty of modeling the complexity of the magnetic field configuration, during the late-ascending phase of the solar cycle, to which CR 2137 belong. It is noted that there were three ICMEs reported in the Richardonson/Cane catalog (Cane and Richardson, 2003; Richardson and Cane, 2010); on May 26th (mean speed 660 km/s), June 6th (mean speed 450 km $\rm s^{-1}$) and June 8th (mean speed 430 km/s) during CR 2137 (2013). These are marked by the red vertical lines in the middle panel in Figure \ireffig:swspred. The solar wind speeds before the start of ICMEs were about 600, 470 and 450 km $\rm s^{-1}$, respectively. These values are hourly averages from the OMNI data of in situ solar wind measurements but the results we presented here are the daily averages; this, along with the fact that the mean speed of the ICMEs were comparable to the solar wind speed before and after the CME passage, implies that most of the effects of the CME must have been smoothed out. Despite this, the model seem to have reproduced the solar wind modulations reasonably.

Figure \ireffig:swserr shows the uncertainty estimates for the 98 Monte-Carlo realizations for the three Carrington rotations selected for the present study. Here, the RMS errors between observed and predicted $\rm{V}_{\rm{sw}}$ using the CSSS model for CR 2102 is represented by the black dashed line, CR 2137 by the red dashed line and CR 2160 by the blue dotted line. The horizontal solid lines and the numbers within the brackets on the right hand top corner depict the mean RMS for the respective Carrington rotations indicated. It is noted that the errors, in general, tend to be larger during solar maximum as evident from the larger values seen for CR 2137. As clear from the figure, there is a large scatter for the RMS error for all the CRs studied and suggests that the errors in the arrival times of solar wind and other solar disturbances (e. g. CMEs) are largely influenced by the errors (standard deviation) in the synoptic maps.

sec:discussion

The standard deviation maps (Figure \ireffig:hmi_maps) represent the uncertainties in the preparation of the photospheric magnetic flux density synoptic maps from the magnetograms and do not represent the errors in the measurements of the magnetic field. This uncertainty is in addition to all other contributions such as instrument noise and magnetic flux imbalance. Our aim in this paper is not to provide a method or a model with better predictive capability but to present a confidence level or an uncertainty estimate in the predicted $\rm{V}_{\rm{sw}}$ based on the uncertainties in the synoptic map used as boundary data to the model that predicted the $\rm{V}_{\rm{sw}}$. This information (the uncertainty estimate), along with their origin (or source) is critical for devising methods to improve the prediction accuracies. Given that such information (uncertainty estimates) is not usually provided when the synoptic maps are produced by various observatories (ground-based and spacecraft measurements) or when solar wind prediction is carried out, the relevance of the uncertainty estimate becomes all the more significant. Our aim, in this paper, is to convey this point by demonstrating how the errors (or uncertainties) in the boundary data (the synoptic maps) used in the models for solar prediction propagates and influence the accuracy of these predictions, and to emphasize how important it is to incorporate this information (the uncertainties) into future efforts to improve the prediction accuracy.

The photospheric magnetic flux density synoptic maps have been produced and used for decades for interpreting the various solar phenomena and observations. It is well established that the solar wind prediction is highly sensitive to the quality of these synoptic maps (e.g. Arge and Pizzo, 2000; Arge et al., 2010; Riley et al., 2014) that are used as the inner boundary conditions of coronal models based on which the solar wind prediction have been made. Our use of magnetic vector field measurements gives us genuine observations of the radial flux distribution, which is not the case with the longitudinal field measurements usually employed in global coronal/heliospheric modeling. However, an estimate of uncertainties in the construction of these maps have never been provided until Bertello et al. (2014) produced the spatial standard deviation maps (§ \irefsec:method). Similarly, a comprehensive estimate of uncertainties in the predictions of $\rm{V}_{\rm{sw}}$ & HMF, locations of CH & NL and other solar wind properties are also unavailable. Lack of such estimates results in poor understanding of the causes and incorrect identification of the sources of the discrepancies between predictions and observations, and thereby, inadequate mitigation of these factors.

In this paper, we obtained systematic and reliable estimates of uncertainties of the coronal and solar wind properties predicted using the HMI photospheric flux density synoptic maps and the corresponding spatial standard deviation synoptic maps (Bertello et al., 2014). For this, we computed the locations of the CHs (photospheric footpoints of open field regions) and NLs, and the FTEs and $\rm{V}_{\rm{sw}}$ using the Current Sheet Source Surface (CSSS) model (Zhao and Hoeksema, 1995; Poduval and Zhao, 2014; Poduval, 2016) of the corona which takes the synoptic map as the inner boundary condition. We carried out the analysis for three Carrington rotations CRs 2102 (3 – 30 October, 2010), 2137 (14 May – 11 June, 2013) and 2160 (1 – 28 February, 2015), representing the different phases of the solar cycle. We compared the locations of CHs and NLs with the corresponding locations in the EUV synoptic maps for the same periods and those computed by the well–established PFSS model. The models and observations exhibit a close match, in general, as seen in Figures \ireffig:chnl1 – \ireffig:chnl3. A quantitative comparison of the $\rm{V}_{\rm{sw}}$ predicted by the CSSS model for these Carrington rotations with the correspsonding in situ observations of solar wind taken from the OMNI database was made by obtaining the RMS error as shown in Figure \ireffig:swserr. We noted that there is considerable spread in the predicted $\rm{V}_{\rm{sw}}$ as reflected in the RMS errors over the different standard deviation maps (MCRs) for a given Carrington rotation. Moreover, the RMS errors are larger during CR 2137, a period during the solar maximum phase – this is mainly due to the difficulty in modeling the complex magnetic field configuration during solar maximum as expected. The uncertainty in the solar wind prediction, on average, based on Figure \ireffig:swserr, varied between 88 and 110 km/s, indicating a significant ambiguity between the slow and fast winds which has serious implications in space weather forecast.

While the uncertainties originating due to the lack of far side information of the Sun, limitations of the model used calibration errors in the synoptic map construction, and the uncertainty in the propagation and arrival time of CMEs, a few to mention, are still significant factors in determining the prediction accuracy, the present analysis points out that the spread in the results, as shown here, is due to the spread in the boundary data values in the Monte Carlo simulations (especially at the poles).

The coronal models, the empirical relationship for solar wind prediction and the kinematic approach for the forward propagation of solar wind we employed here are simple and computationally inexpensive¹¹1For the present work, the solar wind predictions using the 98 MCRs for each of the three Carrington rotations costed us about 100 days of CPU time (§ \irefsec:data).but they form the basis of the current solar wind prediction schemes such as the operational WSA model. Therefore, though our model represent a static corona and does not handle transients (as already known to the scientific community), our results can be directly compared with those of the state-of-the-art model and other sophisticated space weather forecast models such as Enlil. Further, the present work indicates that (Figure \ireffig:swspred) with appropriate corrections to the photospheric synoptic maps, the accuracy of solar wind prediction can be improved significantly. Moreover, the results presented here indicate the importance of “ensemble forecast” in improving the space weather forecast, as the results indicate that a suitable combination of the top performing models (lower values of RMSEs) can provide better and more accurate solar wind prediction.

In order to obtain a better statistics and generalize our findings, we intent to expand the current study over longer periods of time. Since the near–Sun observations of Parker Solar Probe have already been released to the public, our near–Sun predictions can be better validated for optimizing the model performance, and, thereby, improve our near–Earth predictions as well.