 LETTER
 Open Access
Main field and secular variation candidate models for the 12th IGRF generation after 10 months of Swarm measurements
 Diana Saturnino^{1}Email author,
 Benoit Langlais^{1},
 François Civet^{1},
 Erwan Thébault^{1} and
 Mioara Mandea^{2}
 Received: 12 February 2015
 Accepted: 3 June 2015
 Published: 20 June 2015
Abstract
We describe the main field and secular variation candidate models for the 12th generation of the International Geomagnetic Reference Field model. These two models are derived from the same parent model, in which the main field is extrapolated to epoch 2015.0 using its associated secular variation. The parent model is exclusively based on measurements acquired by the European Space Agency Swarm mission between its launch on 11/22/2013 and 09/18/2014. It is computed up to spherical harmonic degree and order 25 for the main field, 13 for the secular variation, and 2 for the external field. A selection on local time rather than on true illumination of the spacecraft was chosen in order to keep more measurements. Data selection based on geomagnetic indices was used to minimize the external field contributions. Measurements were screened and outliers were carefully removed. The model uses magnetic field intensity measurements at all latitudes and magnetic field vector measurements equatorward of 50° absolute quasidipole magnetic latitude. A second model using only the vertical component of the measured magnetic field and the total intensity was computed. This companion model offers a slightly better fit to the measurements. These two models are compared and discussed.We discuss in particular the quality of the model which does not use the full vector measurements and underline that this approach may be used when only partial directional information is known. The candidate models and their associated companion models are retrospectively compared to the adopted IGRF which allows us to criticize our own choices.
Keywords
 Magnetic field
 Main field
 Secular variation
 Modeling
 IGRF
 Time extrapolation
Findings
Introduction
The International Geomagnetic Reference Field (IGRF) is a time series of Main Field (MF) Spherical Harmonic (SH) Gauss coefficients aiming to describe the largescale Earth’s magnetic field of internal origin, also known as the main field. It is published every 5 years and includes a predictive Secular Variation (SV) part for the next 5year period. IGRF models result from a collective and international effort, in order to derive the most accurate model of the main geomagnetic field at a given epoch.
Since the ninth generation of IGRF (Macmillan et al. 2003) Gauss coefficients are computed up to SH degree and order 13 for the static part and up to SH degree and order 8 for the secular variation part. All coefficients are rounded at 0.1 nT or 0.1 nT.yr ^{−1}, respectively.
The latest 12th generation of the IGRF model comes almost 1 year after the successful launch of the ESA threesatellite Swarm mission on 22 November 2013. A full presentation of the mission and of some of its expected outputs can be found in Olsen et al. (2013), Chulliat et al. (2013), and Thébault et al. (2013). After an initial stage where all three satellites flew around 495 km, two satellites fly almost sidebyside at a nominal altitude close to 465 km, while the third one flies some 50 km higher. All three are on near polar orbits. Each satellite carries two magnetic field instruments on a boom. The first one is the Vector Fluxgate Magnetometer (VFM) and is comounted on an optical bench with the Star TRacker (STR) with three Camera Head Units (CHUs) to determine the attitude of the spacecraft. This is necessary to transform the vector readings into geocentric B _{ X }, B _{ Y }, and B _{ Z } magnetic field components (horizontal northward, horizontal eastward, and vertical downward, respectively). The second one is the Absolute Scalar Magnetometer (ASM) and aims at providing very accurate 1 Hz absolute scalar measurements F for both scientific and VFM calibration purposes.
Our candidate model exclusively relies on the measurements made by the lowaltitude Swarm A and C spacecrafts. In the following, we describe the data selection scheme. Because some discrepancies were observed between the scalar magnitude as computed from the VFM measurements and the ASM direct measurements, two datasets were built. In the first dataset, all VFM and ASM measurements were considered. In the second one, we disregarded the horizontal magnetic field components of the VFM measurements. These two datasets are used to derive two models, which are denoted VASM and ZASM, respectively. In the third section, we briefly describe the model parametrization, and compare and discuss the two models in “Comparison of VASM and ZASM models” section, justifying our decision to present the VASM model as our IGRF12 candidate model. Finally, we retrospectively compare our models to the adopted IGRF12 model, which allows us to underline the shortcomings of the chosen approach.
Data selection

flags _B: 0 or 1 (VFM is nominal or ASM is turned off);

flags _F: 0 or 1 (ASM is nominal or running in vector mode);

flags _q: between 0 and 6, or between 16 and 22 (at least two CHUs nominal);

flags _Platform: 0 or 1 (nominal telemetry or thrusters not activated).

5 ≤ Dst ≤ 5 nT for the considered time;

 dDst/dt ≤ 3 nT.h ^{−1} ;

0 ^{0} ≤ Kp ≤ 1 ^{+};

Kp ≤ 2 ^{−} for the previous and following 3h time intervals;

local time between 20:00 and 4:00.
This latter selection criterion is preferred over a more strict one based on the illumination of the spacecraft. This would result in large gaps over polar areas during the summer of each hemisphere (Lesur et al. 2010). VFM and ASM measurements are used within ± 50° quasidipole magnetic latitude, while only scalar measurements by the ASM are considered in the polar areas. Known differences exist between intensity F measurements by the ASM and intensity B computed from VFM measurements, with a root mean square (rms) difference of the order of 1 nT. At the time of deriving the model, no official and definitive strategy has been defined, so we do not take these differences into account and do not scale VFM intensity to match ASM measurements. Instead, we overcome this problem by building two datasets. Both use intensity measurements, but while the first one is completed by full vector measurements, in the second one, we consider only the vertical component of the measurements. This means that the second dataset and associated model depend more moderately on these calibration issues.
In a preliminary stage, we also check data for possible outliers, by looking for possible large discrepancies between observations and predictions by a first version of our model. We chose to eliminate all data acquired on the days when such large discrepencies were observed (year–day of year): 2013352 (VFM), 2014084 (ASM and VFM), 2014085 (ASM and VFM), 2014098 (ASM and VFM), 2014099 (ASM and VFM), 2014181 (ASM), 2014182 (ASM), 2014185 (ASM), 2014188 (ASM). Only Swarm C measurements were eliminated in this step. We however note that this selection came only after data selection with respect to flags and indices. In the last stage of our approach, we further reject measurements associated with large residuals, exceeding 15 nT for B _{ Z }, 25 nT for B _{ X } or B _{ Y } for the VFM, and 35 nT for the ASM (these arbitrary values are about five times the final rms difference). This corresponds to remove about 1 % of ASM measurements and 0.2 % of the VFM triplets.
Model parametrization and statistics
While IGRF MF and SV models are published up to SH degree and order 13 and 8, respectively, we computed parent models to higher degree to avoid possible aliasing (e.g., Whaler 1986). The static part of the internal field, described by \({g_{n}^{m}}\), \({h_{n}^{m}}\) Gauss coefficients of degree n and order m, is computed up to SH degree 25 and the secular variation up to 13. Given the short time interval covered by the data (10 months), we assume a constant secular variation and do not consider secular acceleration. The external magnetic field is described by \({q_{n}^{m}}\), \({s_{n}^{m}}\) Gauss coefficients. It is computed up to SH degree 2. A linear dependence with respect to the Dst index for the first degree is also considered with \(\tilde {q_{n}^{m}}\) and \(\tilde {s_{n}^{m}}\), with internal induced counterpart represented by Q _{1}. Internal and external magnetic potentials at spherical coordinates (r,θ,ϕ) are written as (e.g., Langlais et al. 2003):
where a is the Earth’s reference radius (6371.2 km) and Q _{1} is set to 0.27 (Langel and Estes 1985). The inverse problem is linearized and solved using a least square method (Cain et al. 1989). The choice of the initial model has no effect on the final result as long as it is close enough to the actual field, such as a model at a different epoch (e.g., Langlais et al. 2003). Convergence was reached after two iterations.
There are 881 coefficients to solve, using 38,437 Swarm A ASM scalar measurements, 22,320 Swarm A VFM vector triplets, 40,609 Swarm C ASM scalar measurements, and 21,292 Swarm C VFM vector triplets. The mean epoch of measurements is 2014.3. To overcome the denser data distribution close to the poles, we used a 1/ sinθ weighting scheme (with θ being the colatitude). In the first model, we observed that the misfit for Swarm C was slightly larger than for Swarm A. Because both satellites essentially measure the same magnetic field, they should be associated with similar errors. We therefore chose to give more importance to the latter, with a 9/8 ratio, and weighted the data accordingly.
Root mean square and mean differences (in nT) for the two parent models and for Swarm A and C. The B misfit corresponds to intensity rms difference computed from the VFM dataset. F misfits are sorted with respect to the magnetic absolute latitude 50°
Root mean square difference  Mean difference  

Model  Sat.  B _{ X }  B _{ Y }  B _{ Z }  B  F _{≤50}  F _{>50}  B _{ X }  B _{ Y }  B _{ Z }  B  F _{≤50}  F _{>50} 
VASM  A  4.10  3.94  2.71  3.05  3.07  8.93  0.12  0.72  0.16  −0.09  0.01  −0.46 
VASM  C  4.19  4.49  3.10  3.03  3.11  9.31  0.32  1.41  −0.07  0.19  0.18  −0.26 
ZASM  A  4.38  4.14  2.51  3.04  3.05  8.91  0.58  0.74  0.23  −0.03  0.04  −0.14 
ZASM  C  4.46  4.72  2.88  3.02  3.09  9.29  0.80  1.49  −0.04  −0.33  0.01  0.20 
It is not possible to model the Earth’s magnetic field using only scalar measurements without any prior because of the socalled Backus effect. This effect comes from the nonuniqueness of the inverse problem and is characterized by focused large errors perpendicular to the measured field. This occurs mostly in the equatorial region, and it results in large differences in the vertical component. This effect was discovered and described when no spacecraft vector magnetic field measurements were available (e.g., Backus 1970; Hurwitz and Knapp 1974; Lowes 1975; Stern and Bredekamp 1975). Different strategies have been proposed to alleviate it. Hurwitz and Knapp (1974) were probably the first to include vector data in the equatorial region, to better constrain the position of the magnetic equator and resolve the sectoral harmonics. These additional data can be provided by the magnetic observatories, which have however a poor geographic distribution. Additional information can also be obtained from a triaxial magnetometer on board a satellite, which requires an accurate determination of the satellite attitude (Holme 2000; Holme and Bloxham 1995). Indeed, (Khokhlov et al. 1997, 1999) showed that it is possible to eliminate the Backus effect if the position of the geomagnetic equator (where B _{ Z }= 0) is known. This position can be directly estimated by a time extrapolation from a previous or later epoch model (UltréGuérard et al. 1998a,b) or indirectly from measurements of the equatorial electrojet (Holme et al. 2005).
An approach similar to that of UltréGuérard et al. (1998a) was already employed in the context of IGRF modeling, but this was to test the quality of the candidate models rather than to propose a new model (Mandea and Langlais 2000). Here, we combine direct measurements of the position of the geomagnetic equator (i.e., vertical field measurements) to scalar measurements. The new model will not depend on the possibly more perturbed horizontal components (Table 1), and mismatch between B and F (below 50° absolute magnetic latitude) should not introduce any intrinsic error. This latter point is however debatable, as even the intensity of the measured vertical field depends on the measured F ASM value through the calibration process.
We give in Table 1 the statistics of this second model, denoted ZASM, derived using the second dataset. The rms difference for the B _{ Z } component is improved, with a decrease of about 7 % for both satellites with respect to the VASM model. The misfit for F and B also display a slight decrease with respect to the VASM model. On the contrary, the rms differences for horizontal components and for both satellites are degraded, in a similar proportion than for the B _{ Z } improvement. The mean deviation for B _{ X } difference changes significantly from the VASM to the ZASM model, with an increase of 0.5 nT for both Swarm A and C datasets. A similar change is also observed for F in polar areas.
Comparison of VASM and ZASM models
Root mean square differences at the surface of the Earth (in nT) between the candidate models for different truncation degrees and different epochs. In the last row, only the SV is considered (in nT.yr ^{−1})
Model 1  Model 2  Epoch  Degree  B _{ X }  B _{ Y }  B _{ Z }  B 

VASM  ZASM  2014.3  25  1.39  1.81  1.90  1.48 
VASM  ZASM  2014.3  13  0.72  0.68  1.13  0.94 
VASM  ZASM  2015.0  25  2.71  2.34  3.67  3.26 
VASM  ZASM  2015.0  13  2.46  1.62  3.36  3.04 
IGRF12  VASM  2015.0  13  6.40  5.50  9.22  9.30 
IGRF12  ZASM  2015.0  13  6.77  5.87  9.77  9.90 
VASM  ZASM  2015.0  8 (SV)  2.64  1.65  3.59  2.98 
Comparison with the IGRF12 model
We now compare our candidate and our test models (which are truncated and extrapolated versions of the VASM and ZASM parent models, respectively) to the adopted 12th IGRF generation. This a posteriori comparison is only possible because IGRF was adopted between the time at which we computed our candidate models and the time at which this study is written (Thébault et al. 2015b). Note that IGRF models depend, among others, on our candidate models.
Concluding remarks
We present two candidate models for IGRF12 for the main field at epoch 2015.0 and for the secular variation between 2015 and 2020. We choose to compute parent models with a simple parametrization and without adding regularization or temporal splines. Only Swarm A and C measurements, acquired during the first 10 months of the mission, are considered, with external activity indices selection and outliers removal. We compare two different modeling strategies, one using full vector measurements and one using only vertical component measurements, both in addition to intensity measurements. We show that the differences between these models are small when they are compared at the mean epoch of measurements for n ≤ 13. However, they become larger when the models are extrapolated to 2015.0, increasing from 0.94 to 3.04 nT. This is very likely a consequence of using a too short time interval to construct our SV model.
The two models are relatively similar for the static part, and only the timevarying part is different. The analysis of this difference lead us to chose the VASM parent model of our MF and SV candidates for IGRF. We believe that this difference is related both to a nonoptimal data selection above polar areas (where the misfit is very large) and to a too short time interval to constrain the secular variation. We however want to underline that using the vertical magnetic field in complement to globally distributed scalar measurements to reduce the Backus effect is promising, and that such approach may be explored in the future if required.
Declarations
Acknowledgements
The authors acknowledge ESA for provided access to the Swarm L1b data. DS and FC are supported by CNES and Région Pays de la Loire. This work was partly funded by the Centre National des Études Spatiales (CNES) within the context of the “Travaux préparatoires et exploitation de la mission Swarm” project. We also thank two anonymous reviewers for their detailed comments.
Authors’ Affiliations
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