The Polar Cap (PC) Index: PCS Version Based on Dome-C Data

The standard polar cap (PC) indices, PCN (North) based on magnetic data from Qaanaaq in Greenland and PCS (South) based on data from Vostok in Antarctica, have been submitted from the Arctic and Antarctic Research Institute in St. Petersburg, Russia, the Danish Meteorological Institute, and the Danish Space Research Institute in different versions. In order to consolidate PCS indices based on Vostok data or replace poor or missing index data, derivation procedures have been developed to generate alternative PCS index values based on data from Dome Concordia (Dome-C) magnetic observations from epoch 2009–2020 of solar cycle 24. The


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The present versions of the polar cap (PC) index are based on the formulation by Troshichev et al. (1988) for the version developed jointly by the Arctic and Antarctic Research Institute (AARI) and the Danish Meteorological Institute (DMI). The new idea was the scaling on a statistical basis of the ground magnetic variations to the merging electric field (energy coupling function), E M , based on solar wind parameters (Kan & Lee, 1979) in order to make the PC indices independent of local ionospheric properties and their daily and seasonal variations. Furthermore, for the scaling of PC index values they used components of the magnetic variations in an "optimal direction" assumed being perpendicular to the average DP2 transpolar convection in order to make the new index focused on solar wind-magnetosphere interactions.
The standard PC indices, PCN (North) and PCS (South) are derived from polar magnetic variations recorded at Qaanaaq (Thule) in Greenland and Vostok in Antarctica, respectively. The formulation of derivation procedures has taken three directions related to the contributions by Vennerstrøm (1991), Troshichev et al. (2006), and Stauning et al. (2006). The PCN and PCS versions developed at the DMI by Stauning et al. (2006) and Stauning (2016) are modifications of the Troshichev et al. (1988Troshichev et al. ( , 2006 index versions. The Vennerstrøm (1991) version was abandoned in 2015. A comprehensive description of different PC index versions is available in Stauning (2013b).
The PCN and PCS indices have been used in various versions and combinations in studies of the relations between polar cap disturbances and further activity parameters such as magnetospheric storm and substorm indices. Thus, single-pole PC indices, particularly PCN indices, have been used most widely, but also averages of PCN and PCS indices and seasonal selections (summer or winter) of indices have been used, occasionally just named "PC index," in scientific contributions For the relations between single-pole PC indices and solar wind conditions or global magnetic disturbances there are two conceptual problems. One is the choice between the two available hemispherical indices to be used in such relations. The other is the interpretation of negative index values which could not relate directly to the inherently positive E M values. The combination of non-negative values of PCN and PCS indices introduced by Stauning (2007) and named PCC index have helped solving both problems and underlines the need for alternative PC index data sources to ensure availability of both PCN and PCS indices.
The present contribution presents the potential source for PCS index values in the magnetic data from Dome Concordia (Dome-C) observatory in Antarctica (Chambodut et al., 2009;Di Mauro et al., 2014) in order to enhance the reliability and availability of PCS and PCC indices to be used for solar-terrestrial sciences as well as for space weather monitoring applications. The suggestion to use data from Dome-C (DMC) for an alternative PCS index was initially forwarded in Stauning (2018b). The description of the DMC-based PCS indices and the definition of reference levels and scaling parameters are very similar to the corresponding definitions and descriptions of Qaanaaq (THL)-based PCN indices or Vostok-based PCS indices available in Stauning (2016). An extended description of the index derivation methods beyond the present work may be found in Supporting Information S1 where the disagreements with features of the methodologies endorsed by the International Association for Geomagnetism and Aeronomy (IAGA) are also discussed. Such critical discussions may also be found, among others, in Stauning (2013aStauning ( , 2015Stauning ( , 2018aStauning ( , 2020Stauning ( , 2021aStauning ( , and 2021b).

Basic Principles for Calculation of Polar Cap Indices
The transpolar (noon to midnight) convection of plasma and magnetic fields driven by the interaction of the solar wind with the magnetosphere is associated with electric (equivalent Hall-type) currents in the upper atmosphere in opposite directions of the flow. These currents, in turn, induce magnetic variations at ground level (Troshichev et al., 1988Vennerstrøm, 1991) from which the PC indices are derived.
The PC indices are geomagnetic activity parameters based on polar magnetic variations and scaled to emulate the solar wind merging electric field E M (also termed E KL ), formulated by Kan and Lee (1979) from which the agreed dimension (mV/m) is adapted. In their formulation the merging electric field, E M (V SW ⋅B SW ⋅sin 2 (θ/2)), acting along an assumed merging region corresponding to the projection of the open polar cap to the magnetopause, generates the potential patterns that accounts for supply of energy to magnetospheric disturbance processes and drives the polar cap plasma convection system.
In spite of its dimension in [mV/m], the merging electric field is not a simple measure of interplanetary electric fields. The E M (or E KL ) parameter is a convenient and simplified representation of the solar wind velocity and IMF B Z and B Y parameters of primary importance for the transfer (coupling) of solar wind energy to the magnetosphere providing power to disturbance processes such as polar and auroral current systems, substorm processes, upper atmosphere heating, and building of magnetospheric ring currents.
The steps in the calculations of PC indices may be found elsewhere, for instance in Troshichev et al. (2006) or Stauning et al. (2006Stauning et al. ( , 2016Stauning et al. ( , 2018bStauning et al. ( , 2018cStauning et al. ( , 2020. They are summarized here for convenience and further specified in the associated SI file. In order to focus on solar wind effects, the horizontal magnetic variations, ΔF = F− F RL , of the recorded horizontal magnetic field vector series, F, with respect to an undisturbed reference level, F RL , are projected to an "optimum direction" in space to provide the projected variations, ΔF PROJ . The optimum direction is assumed perpendicular to the DP2 transpolar convection-related sunward equivalent currents and characterized by its angle, φ, with the dawn-dusk meridian. The solar wind energy coupling function, here named "merging electric field," E M , because of its origin and inherent dimension [mV/m] is defined by the expression in Equation 1 (from Kan & Lee, 1979): where V SW is the solar wind velocity, B Y and B Z are geocentric solar-magnetosphere (GSM) components of the IMF, while θ is the polar angle of the transverse IMF vector. The merging electric field is supposed to control the rate of merging (coupling) between solar wind and geospace magnetic fields at the front of the magnetosphere and thereby in control of the input of solar wind energy to the Earth's magnetosphere.
In consequence, the projected polar cap magnetic disturbances, ΔF PROJ , are assumed being proportional to E M : where α is the slope and β the intercept parameter named from a graphical display of the relation.
The PC index is now defined by equivalence with E M in the inverse relation of Equation 2, that is: With the relation in Equation 3, the ΔF PROJ scalar values are scaled to make the PC index equal (on the average) to values of E M in the solar wind. The scaling of the polar cap magnetic disturbances to a quantity in the solar wind removes (in principle) the dependence on the daily and seasonally varying ionospheric conductivities and other local conditions such as the location of the measuring polar magnetic observatory.

Handling of Geomagnetic Observations
The It is evident from Figure 1b that the definition of proper baseline values for Vostok present challenges. The base levels need comprehensive adjustments to remove irregular base level changes and retain secular variations only. Such adjustments are described (to some length) in Stauning (2016). The problem and possible base level corrections are not discussed at all in available reports from the IAGA-endorsed PC index providers at AARI and the DTU Space (e.g., Matzka, 2014;Troshichev, 2011Troshichev, , 2017Troshichev & Janzhura, 2012). The base level problems and occasional missing data supply from Vostok observatory underline the need for alternative PCS index sources.
Corresponding data from DMC observatory are displayed in Figure 2a. In these data there are obvious base level differences between the definitive data during 2016-2017 and the remaining provisional data. However, for DMC data the adjustments are simple and the data quality is otherwise very good. The monthly and yearly average data values after level corrections are displayed in Figure 2b.

Reference Level (QDC) for PC Index Calculations in the SRW Version
The definition of reference levels, F RL , to be used for calculations of the polar magnetic variations needed for PC index calculations differs among the PC index versions. In the version developed at AARI, the varying level on "extremely quiescent days"  was used as the data reference level. This level could be considered built from a quiet day curve (QDC), F QDC , added on top of the base level, F BL . Thus, in vector formulation: Extremely quiescent days are particularly rare at polar latitudes. Therefore, the concept was broadened to imply the generation of QDC values from quiet segments of nearby days within 30 days at a time Troshichev et al., 2006). The use of an interval close to the solar rotation period (∼27.4 days) with equal weight on each day's quiet samples removes most solar rotation effects from the QDCs.
The definition of the reference level is one of the issues that distinguish the PC index version presented in Stauning (2016) and used in the present work from the IAGA-endorsed PC index versions. The reference level construction used here (Equation 4) is based on the formulation in Troshichev et al. (2006) but uses the "solar rotation weighted" (SRW) QDC construction published in Stauning (2011) instead of the 30-days equal weight QDC methods detailed in Janzhura and Troshichev (2008) or the version with the added solar sector (SS) term detailed in Janzhura and Troshichev (2011), Matzka and Troshichev (2014), and Nielsen and Willer (2019).
As formulated in Stauning ( , 2020, the essential point for the SRW method is deriving the reference level from quiet samples collected on nearby days at conditions otherwise as close as possible to those prevailing at the day of interest. Weight functions are defined to optimize the effects on the QDCs with respect to sample separation and solar rotation (see details in the SI file). For each hour of the day, observed hourly average values at corresponding hours within an extended interval (±40 days) are multiplied by the relevant weights, added and then divided by the sum of weights to provide hourly QDC value. Subsequently, the hourly QDC values are smoothed to remove irregular fluctuations and interpolated to provide any more detailed resolution as required.
The derived QDCs are routinely displayed in yearly plots for each component like the example shown in Figure 3.
In these diagrams for the magnetic data from DMC there is a QDC curve for each day of the year. For 1 month at a time, the daily QDC curves are drawn on top of each other in blue line. For day 1 (in black line), day 15 (yellow), and last day of the month (in red line) the QDCs are re-drawn on top of the other QDCs. Going from the black through the yellow to the red curves provides an impression of the development of the QDCs throughout the month. The seasonal variations are very distinct with amplitude maxima at local summer. Most of the additional variability in the QDCs is caused by the IMF B Y -related solar sector effects which are taken into account this way.
The weighting over ±40 days makes the determination of the final QDCs fairly insensitive to intervals of missing data. Thus, the weighting technique allows calculations of real-time QDCs with reduced accuracy from past data collected within −40 to 0 days (actual time) by simply ignoring the not yet available post-event samples without changing the ±40 days' calculation scheme. As further data arrive, then the QDCs could be gradually improved

Optimum Angle Calculations
At the correlation studies by Stauning (2016) using 5-min samples, the best correlations between OMNI Bow Shock Nose values of E M and Qaanaaq ground-based ΔF PROJ data series were obtained for delays close to 20 min.
With the delay fixed, the optimum direction angles are now derived by the method defined in Stauning (2016). For each calendar month and each UT hour of the day and with steps of 10° in the optimum direction angle through all possible directions, the disturbance vectors, ΔF, are projected to the optimum direction while the correlations between the projected magnetic disturbances and the solar wind merging electric fields are calculated using standard Pearson's product-momentum formula.
Among the calculated values of the correlation coefficients derived through all steps in optimum direction angle, the maximum value is found. Based on the direction angle for this maximum value along with the angles for the preceding and the following values of the correlation coefficient, a parabolic function is then adapted to determine the precise values of the optimum direction angle at the top of the parabola and the corresponding maximum correlation coefficient for the calendar month and UT hour in question.
In order to make the values generally representative some averaging and smoothing is necessary. In the present version, the values are exposed to bivariate Gaussian smoothing over months and UT hours by weighted averaging. The exponents used in the smoothing weight functions characterize the degree of smoothing and are stored with the derived optimum direction values. The resulting mean hourly optimum angles for cases without QDC adjustments and excluding NBZ reverse convection samples (blue line), with QDC and without NBZ samples (magenta line with dots), and with QDC and including NBZ samples (red line) are displayed for each calendar month in Figure 4.

Calculations of Slope and Intercept
Recalling that we are searching for proxy values based on polar magnetic disturbances to represent the solar wind "merging" electric field (E M = E KL = V SW B T sin 2 θ/2), the general assumption is that there is a (statistical) linear relation between the polar magnetic variations, ΔF PROJ , and the solar wind merging electric field (coupling function), E M , and that this relation can be inverted and used to define a polar cap (PC) index by equivalence (cf. Equations 1-3). Contrary to the calculation of the optimum direction, the QDC issue has considerable importance for the calculations of slope and intercept parameters. To solve for the coefficients in the linear relation (ΔF PROJ = α E M + β), standard least squares regression is applied on a comprehensive and representative data base. For each calendar month the hourly values of α and β are formed by processing all 5-min values of E M (t-20 min) and corresponding ΔF PROJ (t) throughout that hour of all days of the month and all years of the selected epoch.
In order to avoid reverse convection cases in the data base used for calculations of PC index coefficients, it is required for each sample that IMF B Z < | IMF B Y | + 3.0 nT. This condition excludes cases where strong northward B Z is the dominant IMF component. A further condition imposed on the selection of data requires that the projected magnetic variation, ΔF PROJ , is larger than the value corresponding to PC = −2 mV/m (≈−50 nT). This condition ensures that cases with strong reverse convection, which may continue for a while after the driving northward IMF parameter has been reduced or has changed polarity, are also omitted.
The raw (non-smoothed) values of the slopes and intercept coefficients are exposed to bivariate Gaussian smoothing over months and UT hours by weighted averaging (Stauning, 2016). The resulting slope and intercept values for epoch 2009-2020 are presented in Figure 5 in the format corresponding to Figure 4. Each of the 12 monthly sections presents the mean hourly variation in the parameters for the (calendar) month. The monthly mean hourly values of the slopes and intercepts are converted into series of hourly values for each (calendar) day of the year by Gaussian bivariate weight function interpolation. For finer resolutions, for example, 5-min or 1-min samples, a simple parabolic or linear interpolation is used (Stauning, 2016). It is seen from Figure 5 that the slope values are little affected whether the data are handled with or without QDC.
The intercept values without QDC involvement (blue line) are increased by an amount representing the projected QDC contribution while including the NBZ samples (red line) has no significant effects on slope or intercept. Due to its proximity to the magnetic pole the amount and the intensities of reverse convection events are minimal at DMC which makes the station an ideal location for supply of data for PCS calculations. The calibration parameters are not invariant to general changes in solar activity or to secular variations in the local polar magnetic configuration, but they are kept invariant over years unless a new index version is implemented.

Calculation of PC Index Values Post Event and in Real Time
With the DMI methods (Stauning, 2016), detailed in the SI file, the scaling parameters (φ, α, β), are derived as monthly mean hourly values and then interpolated to provide tables at finer resolution as required. With the optimum angle values displayed in Figure 4, the slope and intercept values displayed in Figure 5, and the QDC values derived by the solar rotation weighted (SRW) method described in the SI file, it is now possible to calculate PCS index values versus UT time and date. The magnetic variations are derived from the observed values by subtracting base line and QDC values. The projection angle for the projection of the horizontal magnetic variation vector (ΔF X , ΔF Y ), in the (rotating) observatory frame at longitude, λ, to the optimum direction, φ, in space is defined by: using the tabulated optimum angles (φ) while UTh is the UT time at the observatory in hours.
Thus, the projected magnetic variations could be expressed by:

Assessments of PC Index Quality
For a geophysical index offered to the international scientific community and important space weather services, the quality of the post event (definitive) as well as the real-time (prompt) index values is of utmost importance. In spite of this (seemingly) obvious ascertainment, little efforts have been provided on this issue at past and present PC index versions.
The main quality principles were formulated in Troshichev (2011). With a single exception in 2017, the correlation between 15-min E M and DMC based PCS values seen in Figure 6 is higher -at times much higher -than the correlation between E M and the Vostok-based PCS values and consistently much higher than the correlation between E M and the THL-based PCN values throughout the epoch (2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018)(2019)(2020).
The seasonal variations in the correlation between E M and the PC indices are displayed in Figure 7 by the monthly mean correlation coefficients for 15-min samples averaged over the epoch 2009-2020. The line types are the same as those used in Figure 6. The order of southern months has been rearranged to make seasons match.  The main reason for the low correlations during local summer months is the increased occurrence frequencies and enhanced intensities of reverse convection events compared to conditions at (local) winter. In terms of location, such reverse convection events are particularly frequent and intense midway between the Cusp region at the dayside and the geomagnetic pole. Thus, they are less frequent and intense at Vostok compared to Qaanaaq and furthermore less frequent at DMC compared to Vostok due to the closer proximity to the (southern) geomagnetic pole (cf. Table 1).

Examples of Dome-C-Based PCS Indices
The availability of magnetic observations and the derivation of calibration parameters from Dome Concordia data are important for reliable investigations of space weather effects by providing back-up for the PCS index values particularly in cases where the harsh Antarctic environment may inhibit supply of data from Vostok or invalidate data quality. Correspondingly, the supply of data for PCN index values might be consolidated by using alternative sources of magnetic data such as Resolute Bay (RES) in Canada or Thule Air Base (TAB) in Greenland (Stauning, 2018b). An example of PCN and PCS values compiled from these sources is displayed in Figure 8 for the strong magnetic storm (Dst (min) = −222 nT) on 16-19 March 2015.
It is evident from Figure 8 that the main polar geomagnetic activity parameters such as the PCC indices (Stauning, 2007(Stauning, , 2012(Stauning, , 2021c(Stauning, , 2021dStauning et al., 2008) which need available PCN as well as PCS indices could be restored with high confidence from the abundance of index sources even in the absence of a single data source. The future supply of regular magnetic observations from Alert close to the northern geomagnetic pole (cf. Table 1) would make the derived PCN indices still more relevant being void of auroral substorm and NBZ effects.
The PCN and PCS indices could be combined differently to form the dual-pole PCC indices. In the strong and complex magnetic storm on 23-26 July 2015 (Dst (min) = −204 nT) shown in Figure 9, the Qaanaaq-based PCN indices have been combined with the Vostok-based PCS indices to form the PCC indices displayed in blue line The differences between the two alternative PCC indices are just a small fraction of their amplitudes such that either version would suffice for most space weather applications such as estimates of the solar wind energy input or ring current enhancements (Stauning, 2012(Stauning, , 2021a(Stauning, , 2021c.
Furthermore, for space weather monitoring as well as for scientific investigations of solar wind-magnetosphere interactions, the double variety of index versions would provide an insurance against faulty interpretation of the situation caused by invalid data from any single source.

Invalid IAGA-Supported PCS Indices
In spite of IAGA guidance through forming the "Index Endorsement Criteria" (2009) and the PC index endorsement by IAGA Resolution no 3 (2013) and furthermore the involvement in the ISGI, the IAGA-endorsed PC index series are poorly documented and not reliable.
One issue is the reference level construction (Janzhura & Troshichev, 2011;Troshichev & Janzhura, 2012) that may cause unfounded changes in the reference level during several days around any particularly strong disturbance event or cause considerable changes in the night-time reference level from daytime cusp-related disturbances (see Stauning, 2013aStauning, , 2015Stauning, , 2020. Another issue is the statistical handling where the non-linear processing (smoothing) of fluctuating scaling parameters based on small initial batches of data samples generate systematic errors as documented in Stauning (2021b). A further issue is the mixing of DP2 (forward convection) and DP3 (reverse convection) samples in the calculations of scaling parameters (see Stauning, 2015). A particularly alarming issue is the lack of verification of methods and control of the PC index series offered to the scientific community.
A striking example of invalid PCS index values is displayed in Figure 10  It is seen that the daily excursions between −2 and +4 mV/m (magnetic storm level) in the IAGA PCS values (red line) must be in error when compared to the other index values recorded on these rather quiet days. In passing it might be noted that the Vostok-based PCS indices (magenta line) agree well with the DMC-based PCS index values (green) in the DMI versions.
The PCN and PCS index values in the IAGA-supported versions (blue and red lines) were downloaded in September 2021 from the "definitive" version link at the AARI web site https://pcindex.org and confirmed by the identical index data downloaded also in September 2021 from the IAGA-supported ISGI web service at (http://isgi.unistra.fr).
Corresponding features are seen in Figure 11 holding PC index data for 15-18 December 2011. It is obvious that the daily excursions between −1 and +3 mV/m in the IAGA PCS values (red line) must be in error when compared to the other index values recorded on these very quiet days.
The diagram in Figure 11 was initially presented in Stauning (2020Stauning ( , 2021c but has now been redrawn with PCN and PCS index values in the IAGA-supported versions downloaded in September 2021 from the "definitive"  versions link at the AARI web site http://pcindex.org and (again) confirmed by the identical index data from the IAGA-supported ISGI web service at (http://isgi.unistra.fr).
The Vostok data from this interval (from https://intermagnet.org) are good (cf. Figure 1). Thus, the excessive values in the IAGA PCS data must rely on failures in the processing software which have been in use since the IAGA endorsement by Resolution #3 in 2013.
Similar excessive PCS index values published by AARI and ISGI web services were displayed in Figure 8 of Stauning (2018b) and the failures were reported to the index providers and to IAGA. There were no responses from the index providers. In the reply from 21 May 2018 from IAGA Executive Committee the concerns over the invalid PCS index values were dismissed. However, these erroneous PCS index data have been used in a number of publications since 2013 up to now (2021), among others, in those issued from AARI, which now add to the 40 devaluated publications listed in Stauning (2021b) that have used PC indices in versions now known being invalid.

Conclusions
Due to its close proximity to the (southern) geomagnetic pole, the occurrence frequency and intensity of disturbing reverse convection events (NBZ conditions) as well as the amount of interfering substorm activity are at very low levels at the Antarctic research station Dome Concordia (Dome-C) making the location ideal for supply of basic magnetic data for PCS indices.
1. The characteristics of the PCS indices derived from data from DMC have shown that these data have an unprecedented close relation to values of the merging electric field, E M , derived from parameters in the impinging solar wind. 2. It is strongly recommended that available DMC data (since 2009) are processed to form alternative PCS index values made available to provide substitutes for missing or poor PCS values based on data from the standard observatory, Vostok. 3. Alternative DMC-based PCS index values may form reassuring validation when agreeing with the standard PCS indices based on Vostok magnetic data or provide motivation for critical examination of data and processing procedures in cases of disagreements. 4. It is suggested that efforts are invested in making data from DMC available in real-time and that processing procedures like those presented here and in the SI file are established to generate real-time southern Polar Cap (PCS) indices for space weather monitoring. 5. The present work (including its SI file) provides coherent definitions and detailed descriptions of all steps involved in the generation of Polar Cap (PC) index scaling parameters and index values in their post-event and real-time versions. 6. It is disappointing that IAGA upon endorsing the current official PC index versions by its Resolution #3 (2013) has failed to request comprehensive documentation of derivation procedures, proper validation of methods, and effective quality control of published index series supplied to the international scientific community.
Details of the DMC-based PCS index definitions and derivation methods are provided in Supporting Information S1.

Appendix A: APC Index Quality Control
A convenient method to detect irregular indices is by inspecting monthly diagrams as the example shown in Figure A1 In these diagrams one should look for agreement between amplitudes of E M and positive PC index values while negative PC index values should be related to small (but still positive) E M values. Consistent regular variations between positive and negative values are questionable and should be exposed to critical examination.
Such features are most easily spotted during quiet conditions. Note that PC index levels above appr. 1.5 mV/m, according to Troshichev (2011Troshichev ( , 2017 or Troshichev and Janzhura (2012), indicate magnetic storm or substorm conditions. Figures 10 and 11