In the early phase of this study, only Type 1 CSES HPM data were available. These data only cover geographic latitudes between 65° S and 65° N. To be able to build preliminary main field models, it was, therefore, decided to complement this data set with scalar data from the Swarm Alpha satellite. The goal was to test the value of the Type 1 CSES HPM data for such modelling purposes. The strategy we adopted was to focus on a simple modelling strategy only using 2 months of data (August–September 2018) when CSES Type 1 data were available and Swarm Alpha was orbiting at a similar local time, providing high-latitude scalar data distribution roughly mimicking the scalar data distribution CSES Type 2 data could ultimately provide. The data selection and modelling strategy used was kept simple to match the only 2 months data availability, and inspired by standard data selection and modelling strategies, such as that used by Vigneron et al. (2015) and Hulot et al. (2015a).
CSES HPM data selection
Only Type 1 CSES HPM data between August 01 and September 30, 2018 were used. 1 Hz scalar data were taken from the CDSM instrument without any geographic restriction (except for the fact, of course that no Type 1 CSES HPM data were available at high geographic latitudes beyond 65° S and 65° N). Since FGM-S2 was further away from the satellite body (see Fig. 1), Type 1 CSES HPM data from this instrument were initially assumed to be of the best quality (rather than FGM-S1), and thus selected for providing the needed 1 Hz vector data (expressed in the instrument’s reference frame). These were further selected according to Quasi-Dipole (QD) latitude (Richmond 1995), using two alternative choices (for testing purposes). A first selection involved selecting vector data at QD latitudes between − 55° and + 55° (to which we will refer as the 55°QD selection). A second selection involved selecting vector data at QD latitudes between − 20° and + 20° (to which we will refer as the 20°QD selection). To avoid spurious data (due to interference by e.g., the TBB instrument) all vector and scalar data were also screened to ensure that no scalar data (or modulus of the vector data) departed from predictions by the CHAOS-6- × 8 model (latest version of the CHAOS-6 model of Finlay et al. (2016) available at the time) by more than 300 nT. Such pre-screening of data using a reasonable prior model is standard practice (see, e.g., Finlay et al. 2016; Vigneron et al. 2015; Hulot et al. 2015a) to remove the relatively few most obvious outliers without biasing the bulk of the data towards the chosen prior model (the choice of the 300 nT ensuring this). In addition, for both vector and scalar data, only night-side data were used, using classical criteria to avoid perturbations due to external sources (LT between 18:00 and 06:00, Kp < 2 + , RC < 2). Finally, all data were decimated (one point every 2 min) to avoid noise correlation between consecutive data and oversampling along the satellite track, while keeping enough data, given the targeted level of modelling.
Swarm alpha data selection
Swarm Alpha data were used for two different purposes. The first was to provide the scalar data at QD latitudes poleward of ± 55° needed to complement the Type 1 CSES HPM data to be able to produce main field models. This first set of data was selected according to the same criteria as the Type 1 CSES HPM data, further requesting that Em < 10 mV/m, and decimated in the same way.
The second purpose was to provide additional data for building reference models entirely based on Swarm data, over the same August to September 2018 time period, sharing similar local time properties and selection criteria as the CSES data. In addition to the previous high QD latitudes scalar data, two additional Swarm Alpha data sets were thus prepared, including 1 Hz scalar and vector data (expressed in the Swarm Alpha VFM vector field magnetometer reference frame) and selected according to similar criteria as either the 55°QD selection (first data set) or the 20°QD selection criteria (second data set) described above for the CSES HPM data. These data were again decimated in the same way.
Model parameterization and optimization
Model parameterization was chosen to be the same for the four models we derived in this preliminary series of tests (one CSES model and one Swarm model for each 55°QD or 20°QD data selection). This parameterization is a simplified version of that used by Vigneron et al. (2015) and Hulot et al. (2015a). Simplification involved parameterizing the main field only up to spherical harmonic (SH) degree and order 15, and only allowing for a linear secular variation (SV) up to degree and order 5. This maximum degree was chosen to account for the fact that only 2 months of data were considered, and that changes in the field due to higher degree SV during such a short period are below the resolution of the data and cannot be resolved. No special procedure was used to handle the crustal field signal above degree 15 (which is neither modelled, nor removed), since this signal also appears to mainly be beyond recovery with just 2 months of data. To describe the external (magnetospheric) and corresponding Earth-induced fields, we mainly followed the CHAOS-4 model parameterization (Olsen et al. 2014, also used by Hulot et al. 2015a). In practice, however, only simplified parameters to account for the remote magnetospheric sources and the near magnetospheric ring current were included. Using the notation of Olsen et al. (2014, see their Eqs. 4 and 5), remote magnetospheric sources (and their induced counterparts) are thus described by a zonal external field up to degree 2 in geocentric solar magnetospheric (GSM) coordinates (2 coefficients, \({q}_{1}^{0,GSM}\) and \({q}_{2}^{0,GSM}\)), while the near magnetospheric ring current (and its induced counterpart) is described using solar magnetic coordinates (SM, see Hulot et al. 2015b, for definitions of the GSM and SM coordinate systems, and Maus and Luehr 2005 for the justification of such an approach). However, only a static field up to degree 2 (\({\Delta q}_{1}^{0}\), \({\Delta q}_{1}^{1}\), \({\Delta s}_{1}^{1}\), \({q}_{2}^{0}\), \({q}_{2}^{1}\), \({q}_{2}^{2}\), \({s}_{2}^{1}\), \({s}_{2}^{2}\)) and a time-varying part proportional to the RC index for degree 1 (\({\widehat{q}}_{1}^{0}\), \({\widehat{q}}_{1}^{1}\),\({\widehat{s}}_{1}^{1}\)) are assumed, leading to 11 parameters in total. Finally, only one set of Euler angles (assumed static throughout the two months time period considered) was also solved for to recover the unknown rotation between the vector instruments (FGM-S2 for CSES, VFM for Swarm) and the STR data provided by each mission. This choice was intended for potential issues with the stability of this rotation to best manifest themselves in the data residuals (see Figs. 6 and 7 and later discussion). In total, 306 parameters were thus solved for, 255 for the static Gauss coefficients, 35 for the linear SV, 13 parameters for the external field, and 3 for the Euler angles.
For solving the inverse problem, we relied on an iteratively reweighted least-squares algorithm with Huber weights (as in Olsen et al. 2014, see also e.g., Farquharson and Oldenburg 1998). The cost function to minimize is \({e}^{T}{C}^{-1}e\), where \(e={d}_{obs}-{d}_{mod}\) is the difference between the vector of observations \({d}_{obs}\) (in the reference frame of the instrument) and the vector of model predictions \({d}_{mod}\), and \(C\) is the data covariance matrix (updated at each iteration). No regularization was applied, but a geographical weight was introduced, proportional to \(\mathrm{sin}\left(\theta \right)\) (where \(\theta \) is the geographic co-latitude), to balance the geographical sampling of data. Both scalar data and Huber weights make the cost function nonlinearly dependent on the model parameters. The solutions were, therefore, obtained iteratively, using a Newton-type algorithm.
A priori data error standard deviations were set to 2.5 nT for both scalar and vector data in all cases (Swarm and CSES data). Attitude error was assumed isotropic (using the formalism of Holme and Bloxham (1996)). Different values were chosen for CSES (100 arcsecs) and Swarm (10 arcsecs), however. A much higher value was indeed required for CSES to account for the significantly lower quality of the mechanical link between the CSES STR reference frame and FGM reference frame (see below).
Lessons learnt
Four models were produced in total. Two were built using the Type 1 CSES HPM vector and scalar data from either the 55°QD or the 20°QD selection, complemented with high-latitude Swarm Alpha scalar (as described above). For brevity, we will refer to these as the 55°QD and 20°QD CSES models. Two additional Swarm reference models were otherwise built in the same way, using the 55°QD and 20°QD Swarm data selections (55°QD and 20°QD Swarm models). Figures 2 and 3 illustrate the corresponding data distributions. Comparing Figs. 2a, 3a (availability of vector data between − 55° and + 55° QD latitudes from, respectively, CSES and Swarm Alpha) reveals a significant difference between the CSES and Swarm data distributions.
Whereas the Swarm Alpha orbit provides a nice global coverage of all longitudes over the 2 months considered, the 5-day revisiting period of CSES is responsible for a significantly poorer longitudinal distribution, leaving roughly 80 sectorial gaps. However, we note that by the Nyquist sampling criterion, 80 equally spaced bands in longitude should allow the recovery of sectorial dependence up to order 40. These gaps are thus expected to be narrow enough to only mildly affect the recovery of a global field model up to degree and order 15. Indeed, this does not turn out to be the most significant issue.
A much more significant issue is revealed by the comparison of the CSES and Swarm 55°QD models, as shown in Fig. 4. For SH degrees 1 to 4, the Lowes-Mauersberger spatial spectrum (Mauersberger 1956; Lowes 1966) of the differences between these two models at the Earth’s surface for central epoch of the models (September 1, 2018) is clearly much larger than that of the differences between the Swarm 55°QD model and the CHAOS-6- × 8 model for the same epoch. The latter spectrum provides a good indication of the limitation of using only 2 months of 55°QD selected data from a single satellite. Clearly, the CSES 55°QD model fails to properly determine the first four spherical harmonic degrees of the field. Plotting the radial component of the difference between the predictions of the CSES and Swarm 55°QD models at the Earth’s surface (also shown in Fig. 4) makes it clear that this disagreement, reaching up to 70 nT at Earth’s surface, is mainly zonally distributed and not related to the sectorial gaps seen in Fig. 2. Its magnitude also makes it difficult to relate to differences in the magnetic field signals seen by the Swarm and CSES satellites, which share similar altitudes, or to some potentially poorly recovered secular variation, which cannot produce such differences between models built with only 2 months of data. Although one cannot exclude that this disagreement could be due to some other unidentified issue, the most likely possibility we identified is related to the mechanical link between the FGM_S2 instrument (on the last leg of the boom, see Fig. 1) and the STR (providing attitude information, but located on the body of the satellite). This link is prone to potential systematic deformation along the orbit. Recall, indeed, that our modelling procedure assumes this link to be strictly rigid throughout the 2 month period considered, whereas the design of the CSES HPM boom (three segments with three hinges) may not be capable of guaranteeing this.
To check this possibility and attempt to improve the quality of the CSES model to be recovered, we relied on similar comparisons, now using the CSES and Swarm 20°QD models. These models are based on much less vector data, all concentrated in a 40°QD wide equatorial band along the magnetic equator. The hope was that the mechanical link (rotation matrix) between the FGM_S2 and STR frames of reference would be stable enough along this equatorial part of the (night-side) orbit leg, and similar enough from one orbit to the next, to behave as if almost stiff. Ignoring all vector data was obviously not an option, since enough vector data close to the magnetic equator are mandatory, in particular to provide the knowledge of where this equator lies, a critical information (see Khokhlov et al. 1997, 1999) to avoid the recovered model being affected by the so-called Backus effect (Backus 1970, also known as the perpendicular effect, Lowes 1975). Figure 5, to be compared to Fig. 4, shows that this indeed brings improvement. The disagreement between the two CSES and Swarm 20°QD models for degrees 1 to 4 is much reduced. The reduced use of vector data comes at a slight cost, though, with a modest degradation of the recovery of the degree 5 SH component (see also the impact on the Swarm 20°QD model when compared to the CHAOS-6- × 8 model). Overall, nevertheless, the improvement is very substantial, as can also be seen in the map of the radial component of the difference between the predictions of the CSES and Swarm 20°QD models plotted at the Earth’s surface (also shown in Fig. 5). Although the zonal effect is not entirely removed, it now leads to disagreements about three times less in magnitude, only reaching 25 nT at most at Earth’s surface (note the difference in the colour scales used in Figs. 4, 5).
To further confirm that the issue in the CSES models is indeed likely linked to some deformation of the boom along the orbit, we finally computed the residuals between the CSES Type 1 vector data used and the predictions of the CHAOS-6- × 8 model (which includes both internal and magnetospheric source contributions, but not, e.g., in situ ionospheric currents crossed by the satellite). Should the CSES vector data be free of any slowly varying biases (such as produced by orbital boom deformation), these residuals would be expected to only reflect noise in the data and contributions of signals from sources not modelled by CHAOS-6- × 8. In contrast, if boom deformation occurs systematically along the orbit, significant signatures would be expected in the form of slowly varying biases as a function of latitude. Since it is known that no such effect is to be found on Swarm Alpha (see, e.g., Olsen et al. 2015; each Swarm satellite has its VFM rigidly linked to its set of STR on a specially designed optical bench), a simple way to check this is to plot the equivalent residuals between the Swarm Alpha vector data and predictions of the CHAOS-6- × 8 model. Both satellites orbiting at the same local time over the time period considered (therefore sensing similar un-modelled sources), the latter residuals are expected to provide a relevant baseline.
Residuals were computed in both the NEC and instrument frames of reference, taking advantage of the Euler angles computed in the course of producing the 55°QD CSES and Swarm models to convert vector components from one frame to the other. Residuals in the NEC frame were computed using the Euler angles and quaternion information to rotate the vector data from the instruments frame to the NEC frame, before subtracting the predictions of the CHAOS-6- × 8 model (Fig. 6). Residuals in the instruments frame were computed using the quaternion information and Euler angles to rotate the predictions of the CHAOS-6- × 8 model before subtracting these from the vector data (Fig. 7).
As can be seen, no significant bias can be found in the Swarm Alpha residuals, which also display a dispersion of the type expected for Swarm, for the quiet night-time selection used in this study (see, e.g., Olsen et al. 2015). In contrast, strong varying biases can be found in the CSES residuals. These biases are strongest in the high southern latitudes, progressively decrease towards the equator, and are much less marked in the northern hemisphere. This North–South asymmetry, we note, is consistent with a similar asymmetry in the disagreements between the CSES and Swarm models (stronger in the Southern hemisphere than in the Northern hemisphere, recall Figs. 4, 5). Since CSES orbits at a fixed 14h00 LT at descending node, this evolution follows the path of the satellite on its night leg of the orbit, from South to North. It shows that the bias is maximum every time CSES moves away from the Sun at the end of the dayside orbit leg during which the boom has been presumably heated, than starts decreasing as the satellite begins its journey northwards in the dark, allowing the boom to progressively cool down. This thus strongly suggests that the bias signature is indeed related to some thermal boom deformation, which builds up on the dayside leg of the orbit, then thermally relaxes on the night-side leg, settling back to a roughly stable state by the time the satellite reaches the equator on this night side. This evolution also shows that the most problematic CSES vector data are those from the southernmost part of the (night-side) orbit. These data being dismissed in the 20°QD data selection, it naturally explains why the 20°QD CSES model appears to be of much better quality than its 55°QD equivalent.
Last but not least, Figs. 6, 7 also clearly show that the dispersion in the CSES residuals is much larger than that in the Swarm Alpha residuals. It is highly doubtful that this could be the result of different natural un-modelled signals seen by the two satellites. The intrinsic noise level affecting the FGM_S2 measurements (due to the instrument, the satellite and the rest of the payload) having been shown to be roughly comparable to that affecting the Swarm Alpha VFM instrument (Zhou et al. 2019), this, we practically, attributed to the impact of the not-so-stiff boom and possibly also errors in the attitude restitution provided by the STR through the quaternions (though independent checks of these STR data, not reported here, suggest that this source of error is much less significant, except possibly on some specific days, see below). This noise level is the reason we assumed a fairly large error of 100 arcsecs for the attitude when computing CSES models.
A number of important lessons were thus learnt from the above preliminary modelling attempts. One is that the a priori unfavourable 5 days recursive period of CSES, which introduces longitudinal gaps in the data distribution (see Fig. 2), does not appear to be critical for IGRF modelling purposes. Another one, unfortunately much more critical, is that the mechanical link between the FGM (on the last leg of the three hinges boom) and the STR (on the body of the satellite) appears to be problematic. The boom seems to suffer from systematic thermal deformations along the orbit of CSES, which affect the recovery of the attitude of the vector data provided by the FGM. This deformation could be roughly characterized, and the issue appears to mainly affect data from the southernmost part of the night-side leg of the CSES orbits needed for IGRF modelling purposes. Nevertheless, a simple workaround could be found, which consisted in selecting vector data only within a 40°QD band centred on the magnetic equator (the 20°QD selection), and assuming an attitude error of 100 arcsecs in the inversion procedure. The timeline imposed by the IGRF deadline of October 1, 2019 did not allow us to test more advanced strategies, and this is the strategy we therefore used to also produce the CGGM parent model as described below. One significant change we made, however, is that we decided not to use the vector data provided by the FGM_S2 instrument, in favour of the vector data provided by the FGM_S1 instrument. This choice was justified by the fact that this instrument being closer to the satellite (recall Fig. 1), boom deformation can be expected to be slightly attenuated, with the potential drawback of having slightly noisier data (because of the smaller distance to the satellite) being minor, since such noise level has not been identified as the limiting factor.
