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Parent magnetic field models for the IGRF12GFZcandidates
 Vincent Lesur^{1}Email author,
 Martin Rother^{1},
 Ingo Wardinski^{1},
 Reyko Schachtschneider^{1},
 Mohamed Hamoudi^{2} and
 Aude Chambodut^{3}
 Received: 30 January 2015
 Accepted: 21 April 2015
 Published: 10 June 2015
Abstract
We propose candidate models for IGRF12. These models were derived from parent models built from 10 months of Swarm satellite data and 1.5 years of magnetic observatory data. Using the same parameterisation, a magnetic field model was built from a slightly extended satellite data set. As a result of discrepancies between magnetic field intensity measured by the absolute scalar instrument and that calculated from the vector instrument, we recalibrated the satellite data. For the calibration, we assumed that the discrepancies resulted from a small perturbing magnetic field carried by the satellite, with a strength and orientation dependent on the Sun’s position relative to the satellite. Scalar and vector data were reconciled using only a limited number of calibration parameters. The data selection process, followed by the joint modelling of the magnetic field and Euler angles, leads to accurate models of the main field and its secular variation around 2014.0. The obtained secular variation model is compared with models based on CHAMP satellite data. The comparison suggests that pulses of magnetic field acceleration that were observed on short time scales averageout over a decade.
Keywords
 IGRF12
 Swarm satellite mission
 Geomagnetic field models
Background
The International Geomagnetic Reference Field (IGRF) is a geomagnetic field model used for numerous scientific and industrial applications. It is updated every 5 years (Macmillan et al. 2003; Maus et al. 2005; Finlay et al. 2010). The model is determined by a group of geomagnetic field modellers associated with the International Association of Geomagnetism and Aeronomy (IAGA) and built by comparison of different model candidates generated by scientists affiliated to different institutions around the world. A single institution can propose only one series of models. In this short paper, we present the model candidates proposed by scientists of the GFZ German Research Centre for Geosciences, in collaboration with scientists from other institutions.
Three model candidates were provided for the IGRF12: two main field model snapshots, one for 2010 and one for 2015, and a predictive linear secular variation (SV) covering years 2015 to 2020. The main field model for 2015 and the SV model were derived from a parent model built from a combination of Swarm satellites and observatory data. This parent model includes a complex timedependent parameterisation of the core field, a static representation of the lithospheric field, the external fields and their induced counterparts. Weaker signals, such as the field generated by the tidal motion of the oceans, are not modelled. Hereinafter, we do not present the derivation of this parent model but a very similar model built following the same approach and using a slightly longer time series of satellite data that have been recalibrated. In the same way, the parent model of the main field snapshot for 2010 is not described. It has been derived from observatory and CHAMP satellite data but otherwise follows the same model parameterisation as the parent model derived from Swarm data.
The Swarm constellation of satellites was launched in November 2013, but the satellites reached their survey orbits only by midApril 2014. At this early stage of the mission, the data are not yet fully calibrated and a specific data set has been provided by the European Space Agency (ESA) to be used for modelling purposes. However, some difficulties had to be handled in order to use these data. In particular, each satellite is carrying two instruments for magnetic measurements, and a correction had to be applied to explain the observed differences between the calculated magnetic field strength from the vector fluxgate measurements (VFM) and the magnetic total intensity obtained form the absolute scalar measurements (ASM). The first part of the second section of this paper is dedicated to the description of this correction. We made the choice to present here the latest version of the different correction processes we studied and the field model associated with it. It will be shown in the last section of this paper that our IGRF candidates are very similar to the model obtained with this corrected data set. Of course, the parent model of our IGRF candidates also used corrected data, but with a slightly less robust correction process than the one presented below. Outside this recalibration, the processing path used to obtain accurate models of the magnetic field is similar to that of previous models of the magnetic field model series GRIMM (Lesur et al. 2008, 2010; Mandea et al. 2012; Lesur et al. 2015) We note however that, unlike during the CHAMP epoch, only a very short time span of satellite data was available at the end of September 2014 when the IGRF candidates were submitted. Therefore, observatory data had to be used to obtain robust models. Furthermore, specific regularisation processes had to be introduced in order to obtain models of acceptable quality.
The next section described the methods used to obtain the field models. In the first step of the modelling effort, data are selected and processed. This is described in detail in the second subsection. The model parameterisation is explained in the third subsection and the data inversion process in the fourth. The results are presented in the third section and discussed.
Methods
Corrections of the ASMVFM differences

Three scaling values, one for each of the three sensors, in the directions (defined below) E _{1}, E _{2} and E _{3}, respectively. These three scaling parameters are s _{1}, s _{2} and s _{3}.

Three offset values, one for each of the three sensors, in the directions E _{1}, E _{2} and E _{3}, respectively. These three offset parameters are o _{1}, o _{2} and o _{3}.

Three angles, called the nonorthogonality angles, that are calculated to insure that the three magnetic field components are in orthogonal directions. These three angles are a _{12}, a _{23} and a _{31}. They describe deviations from 90° of the angles between the E _{1} E _{2}, E _{2} E _{3} and E _{3} E _{1} sensor directions, respectively – i.e. if a _{12}, a _{23} and a _{31} are zero, then the E _{1}, E _{2} and E _{3} sensor directions are already along orthogonal directions. In the process of estimating these three angles, the E _{1} direction is assumed fixed and is not modified. The reorientation of the obtained orthogonal set of directions relative to the Earthfixed, Earthcentred coordinate system is performed at a later stage of the processing, sometimes simultaneously with the field modelling process.
The VFM sensor E _{1}, E _{2}, and E _{3} directions correspond roughly to the direction perpendicular to the satellite boom oriented down, the direction perpendicular to the boom oriented right relative to the satellite flying direction, and the direction along the boom oriented toward the scalar magnetometer, respectively. From the experience gained during previous satellite missions, it is known that the calibration parameters estimated on ground have to be reestimated in flight. A description of the parameters defined for CHAMP and Ørsted satellites can be found in (Merayo et al. 2000; Olsen et al. 2003; Yin and Lühr 2011; Yin et al. 2013). In the case of Swarm satellites, the nonorthogonality angles and the offsets are not expected to change with time, whereas the scaling is likely to change slowly with time because of the ageing of the magnetometers. Due to the structure and mechanical properties of the magnetometers onboard Swarm satellites, it is expected that the rate of change of these scaling values is the same for the three orthogonal directions.
Offsets, scaling and angles obtained for models with and without Sun dependence
Without Sun dependence  With Sun dependence  

Offsets  
E _{1}  0.9656 nT  0.6768 nT 
E _{2}  −1.8153 nT  −1.8555 nT 
E _{3}  0.1655 nT  0.0696 nT 
Scaling  
MJD = 5053.7  
E _{1}  1.0 + 0.915 10 ^{−4}  1.0 + 0.820 10 ^{−4} 
E _{2}  1.0 + 1.024 10 ^{−4}  1.0 + 0.578 10 ^{−4} 
E _{3}  1.0 − 0.254 10 ^{−4}  1.0 − 0.211 10 ^{−4} 
MJD = 5410.4  
E _{1}  1.0 − 0.090 10 ^{−4}  1.0 − 0.007 10 ^{−4} 
E _{2}  1.0 − 1.456 10 ^{−4}  1.0 − 0.430 10 ^{−4} 
E _{3}  1.0 − 1.176 10 ^{−4}  1.0 − 1.089 10 ^{−4} 
Angles  
a _{12}  0.5524 arc sec  0.8156 arc sec 
a _{23}  0.1884 arc sec  −0.3925 arc sec 
a _{31}  −0.3768 arc sec  −3.158 arc sec 
The three panels of Figure 2 display similar patterns, although the central panel is of the opposite sign. This is consistent with a small magnetic perturbation carried by the satellite, generated in the vicinity of the VFM sensors, and that depends on the Sun’s position. In such a scenario, the central panel where the satellite is in a descending mode on the dayside of the Earth  i.e. flying toward South on the dayside  should have roughly opposite sign anomalies compared to the two other panels where the satellite is flying North on the dayside of the Earth. We also observe that the anomalies differ dependent on the Sun being on one side of the satellite or the other  i.e. if α is smaller or larger than 90°. We note that the maximum perturbation is not observed when the Sun is just above the satellite, but rather slightly on the side. Finally, we see that there are areas, as those circled in black on the central panel in Figure 2, where the anomalies are displaying smallscale structures.
where the \({Y_{l}^{m}} (\alpha,\beta)\) are spherical harmonics (SH). The parameters \({o_{1}^{0}}\), \({o_{2}^{0}}\) and \({o_{3}^{0}}\) are the constant offset values, and \(o_{1}^{l,m}\), \(o_{1}^{l,m}\), \(o_{1}^{l,m}\) are the parameters for the Sun position dependence of the offsets. We choose a maximum SH degree of 30 arbitrarily. Recent experiments have shown that this maximum degree can be reduced (Lars TøffnerClausen, personal communication). In the results presented below, we also used a SH representation for the scaling factor up to SH degree 10, but this can probably be dropped if a sensor temperature dependence is assumed instead. Using this parameterisation of the anomaly, we proceed as before and adjust the set of parameters to minimise the differences between the ASM readings and the strength of the magnetic field observed by the VFM instruments through a leastsquares fit.

A large negative anomaly when the Sun is nearly above and behind satellite A, slightly on the right side of the satellite when looking in the flight direction.

A large positive anomaly, just before the Sun lowers below the horizontal plane of the satellite, on its left side.
The offsets in the E _{2} component also show a relatively large anomaly when the Sun is slightly behind the satellite on its right side when looking in the flight direction. These Sundependent offsets, which are supposed to be independent of time, correct the VFM data so that the anomalies presented in Figure 2 vanish.
Overall, the correction is successful. It can certainly be improved and stabilised, but what is described above was the best model available in December 2014. The differences between ASM and VFM values for the other two satellites are much weaker. Nonetheless, the processing described here leads also to a good fit between ASM and VFM data. With such a correction, the standard deviation of the differences between ASM values and the field strength, as estimated from VFM data, is around 0.17 nT for all three satellites.
Data selection
The magnetic field models were derived from the three Swarm L1b Baseline 0301/0302 satellite data series that have been processed and corrected as described in the previous section. The corrected magnetic field values in the VFM reference frame are rotated in the required reference frame  typically the Earthcentred, Earthfixed, North, East, Center reference frame (NEC), when necessary. We also use observatory hourly means as prepared by Macmillan and Olsen 2013.

Satellite A: from MJD 5078.14 to 5369.87

Satellite B: from MJD 5080.02 to 5369.88

Satellite C: from MJD 5086.00 to 5369.87
The observatory data were available only up to MJD 5219.
The selection criteria used for these data are similar to those used in GRIMM series of models  e.g. Lesur et al. 2010. We recall these criteria for completeness.

Positive value of the Zcomponent of the interplanetary magnetic field (IMF B _{ z }),

Sampling points are separated by 20 s at minimum,

Data are selected at local time between 23 : 00 and 05 : 00, with the Sun below the horizon at 100 km above the Earth’s reference radius (a=6371.2 km),

D s t values should be within ±30 nT and their time derivatives less than 100 nT/day, and

Quality flags set to have accurate satellite positioning and two star cameras operating.

Data are selected at all local times and independently of the Sun position.
We point here to the fact that provisional values of the D s t index are used for selection and modelling. The definitive D s t index values are likely to be different. At the time of data selection, no observatory data were available to define a more suitable selection index (e.g. the VMD index defined in Thomson and Lesur (2007)).
Model parameterisation
where \({Y_{l}^{m}}(\theta,\phi)\) are the Schmidt seminormalised spherical harmonics (SH). θ,ϕ,r and a are the colatitude, longitude, satellite radial position and model reference radius, respectively, in geocentric coordinates. We use the convention that negative orders, m<0, are associated with sin(mϕ) terms whereas null or positive orders, m≥0, are associated with cos(m ϕ) terms.
where the function \({\mathcal {H}}_{j}(X)\) takes the value X in the time interval [t _{ j }:t _{ j+1}] and is zero otherwise. θ _{S}, ϕ _{S} are the colatitudes and longitudes in the SM reference frame. For observatory data, we also coestimate crustal offsets.
The external field parameterisation also consists of independent parts. A slowly varying part of the external field model is parameterised over each 6month time interval by a degree l=1 order m=0 coefficient in the geocentric solar magnetospheric (GSM) system of coordinates, and two coefficients of SH degree l=1 with orders m=0 and m=−1 in the SM system of coordinates. The rapidly varying part of the external field is controlled using the external part of the D s t index  i.e. the E s t  and the IMF B _{ y } time series. Here again, 6month time intervals are used. Four scaling coefficients for the E s t are introduced in each interval in the SM system of coordinates: three for SH degree l=1 and orders m=−1,0,1 and one for SH degree l=2 and order m=0. One scaling coefficient for the IMF B _{ y } is introduced in each time interval for SH degree l=1 and order m=−1 in the SM system of coordinates.
θ _{G}, ϕ _{G} and θ _{S}, ϕ _{S} are the colatitudes and longitudes in GSM and SM reference frames, respectively.
We used independent external field parameterisations for the satellite and observatory data. For the latter, we impose that \(q_{1j}^{0 \, \text {SM}}\) is set to zero to avoid colinearities with the observatory crustal offsets.
Independently of the parameterisation of the magnetic field, we also want to estimate the socalled Euler angles between the VFM orthogonal set of measurements and the star camera reference frame, such that the measured vector field in the VFM reference frame can be mapped into an Earthcentered, Earthfixed reference frame. The latter reference frame is usually the NEC system of reference. We actually only compute corrections of predefined Euler angles, for a series of 30day windows. The parameterisation and algorithm we used are detailed in Rother et al. (2013) and are not repeated here.
Inversion process
to be solved, where d is the data vector, g is the vector of Gauss coefficients defining the model and ε is this part of the data that cannot be explained by our model. We note that for midlatitude data, the Equations 5 have to be rotated into the SM system of coordinates because the data at midlatitudes are selected in that system. However, for the internal part of the model, the data in the direction Z _{SM} are not used as they are strongly contaminated by the external fields. This approach has been used in all models of the GRIMM series. More details are provided in Lesur et al. (2008).
Equation 9 and 11 should be solved simultaneously by leastsquares where the λ _{ i } are scalars that need to be adjusted so that the satellite and observatory data are fit to their expected noise level.
Satellite and observatory data weight parameters, residual means and rms
Nb  σ  k  a  μ  rms  

A  B  C  
Satellite  
X _{SM}  122,286  3.4  3.2  3.2  1.0  0.55  0.01  2.76 
Y _{SM}  122,286  3.3  3.4  3.3  1.0  0.55  0.00  2.87 
Z _{SM}  122,286  5.4  5.4  5.5  1.0  0.55  −1.04  4.75 
X _{HL}  276,916  10.0  10.0  11.0  0.7  0.30  0.56  25.77 
Y _{HL}  276,916  11.3  12.2  6.5  0.7  0.30  0.13  28.14 
Z _{HL}  276,916  11.0  11.0  7.0  0.7  0.30  0.04  13.76 
X _{SM,Euler only}  89,461  3.4  3.2  3.2  1.0  0.55  −0.02  2.94 
Y _{SM,Euler only}  89,461  3.3  3.4  3.3  1.0  0.55  0.01  2.76 
Z _{SM,Euler only}  89,461  5.4  5.4  5.5  1.0  0.55  0.77  5.46 
Observatories  
X _{SM}  31,614  4.5  1.0  0.40  0.27  4.03  
Y _{SM}  31,614  4.2  1.0  0.40  −0.22  3.95  
Z _{SM}  31,614  7.0  1.0  0.40  −0.36  5.43  
X _{HL}  8,007  14.0  0.9  0.30  −4.34  24.59  
Y _{HL}  8,007  7.0  0.9  0.30  1.19  15.56  
Z _{HL}  8,007  18.0  0.9  0.30  0.40  24.12 
where j is the iteration number, and W ^{ j } is a diagonal matrix which elements are \(({w}_{i}^{j})^{2}\), with \({w}_{i}^{j}\) defined by Equation 12.
Once an acceptable solution is obtained, a threshold filtering is applied to get rid of some remaining outliers. Then, the Euler angles are estimated and the inversion process is restarted. The final solution is reached when the Euler angles are not changing significantly from one iteration to the next.
Results and discussion
In Table 2, we present the fit to the data for all the used subsets of data, together with the control parameters σ _{ i }, k _{ i } and a _{ i }. These are specified independently for each of the satellites when necessary. The residual rms for the lowlatitude magnetic field components X _{SM}, Y _{SM} and Z _{SM} data are in the same range as for the GRIMM models based on CHAMP satellite data (see Lesur et al. 2015). The Z _{SM} component is included, as it enters in the Euler angles and external field parameter estimation, but it does not affect the estimation of the fields of internal origin directly. The misfits for the highlatitude data are slightly smaller than those obtained with CHAMP data for similar model parameterisation. It is not clear yet why this occurs. The magnetic data selected for the extended local time window are only used for the Euler angle estimation. Since early evening and late morning data are included, the fit to the Z _{SM} component is slightly degraded. Nonetheless, the fit to these data remains surprisingly good.
The misfits for the observatory data are comparable to those of the satellite data, although slightly degraded for the mid and lowlatitude SM components. The mean of the residuals is relatively high for the observatory X _{HL} component which is an indication of the strongly nonGaussian residual distribution. Overall, accounting for the corrections applied to satellite data and the obligation to use preliminary D s t index values for the selection and modelling, the fit to the data is acceptable.
Conclusions
In the view of providing candidates for the new IGRF12, we processed a combination of Swarm satellite and observatory data. In order to obtain robust models of the main field, we first applied a correction to the satellite data such that the strength of the magnetic field, as measured by the VFM instrument, matches the data measured by the absolute scalar instrument. Our underlying hypothesis is that the discrepancies between these two instruments are closely linked to the position of the Sun relative to the satellite. So far, the corrections obtained with such a hypothesis seem suitable. The results presented here are actually part of a longer term study, made in close collaboration with the European Space Agency and several other European institutions. The ultimate goal is to identify the original source of this perturbative signal such that, firstly, the Swarm satellite data can be corrected to remove that signal and, secondly, a new design can be adopted in future satellite missions to avoid such difficulties. As stated in the text, the model presented here is not optimal but corresponds to our best results available at the end of 2014.
Using these corrected data, we have been able to derive accurate models of the geomagnetic field around epoch 2014.0. Since less than a year of satellite data had been accumulated, the SV model  i.e. the magnetic field linear variation in time  is only accurate for the longest wavelengths (up to SH degree 8). We therefore cannot compare meaningfully maps of the SV at the CMB with those obtained at earlier epochs. Nonetheless, the average acceleration of the magnetic field  i.e. its second time derivative  between 2005 and 2014.0 is remarkably weak at the Earth’s surface compared with the acceleration values obtained during the CHAMP era. This observation strongly supports the view that the observed acceleration peaks, which have been associated with magnetic jerks in 2003, 2007 and 2010, correspond to shortterm disturbances of the field over an otherwise slowly and, most of the time, smoothly evolving magnetic field.
Declarations
Acknowledgements
The authors acknowledge ESA for providing access to the Swarm L1b data. We acknowledge also the institutes and scientists running magnetic observatories which provide data that are essential for magnetic field modelling. IW was supported by the DFG through SPP 1488.
Authors’ Affiliations
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