Inflight scalar calibration and characterisation of the Swarm magnetometry package
 Lars TøffnerClausen^{1}Email authorView ORCID ID profile,
 Vincent Lesur^{2},
 Nils Olsen^{1} and
 Christopher C. Finlay^{1}
Received: 29 January 2016
Accepted: 29 June 2016
Published: 22 July 2016
Abstract
Keywords
Introduction
In November 2013 the European Space Agency (ESA) launched the three Swarm satellites, named Alpha, Bravo, and Charlie, with the objective to provide the best ever survey of the geomagnetic field and its temporal evolution (FriisChristensen et al. 2006). Each spacecraft carries an Absolute Scalar Magnetometer (ASM) for measuring Earth’s magnetic field intensity, a Vector Fluxgate Magnetometer (VFM) measuring the direction and strength of the magnetic field, and a threehead Star TRacker (STR) mounted close to the VFM to obtain the attitude needed to transform the vector readings to an Earthfixed coordinate frame. Time and position are provided by an onboard GPS receiver. The payload also includes instruments to measure plasma and electric field parameters as well as nongravitational acceleration.

the vector readings of the VFM are affected by a disturbance vector field;

the scalar readings of the ASM are much less, if at all, affected.
The purpose of this article is to document the details of inflight calibration of the Swarm magnetometer package, including an empirical determination and removal of the Sundriven vector disturbance field \(\delta \vec {B}_\mathrm {Sun}\), based on a mitigation approach proposed by Vincent Lesur (Lesur et al. 2015).
“Characterisation and calibration with scalar residuals” section describes the parameterisation of the model of the Sundriven disturbance—in following referred to as the characterisation of the disturbance field—and of the calibration of the VFM instrument, by which means determination of its intrinsic scale factors and their dependence on time and temperature, and determination of the sensoraxis nonorthogonalities. We document the adopted Iteratively Reweighted Least Squared (IRLS) estimation approach that includes a truncated singular value decomposition (SVD) approach to solving the inverse problem. The results obtained for Swarm Alpha, based on data covering the period from launch (22 November 2013) until end of June 2015 (i.e. 19 months), are presented in “Results of model estimation for Swarm Alpha ” section. Application of the scheme to data from the satellites Bravo and Charlie resulted in similar levels of residual improvement and statistics, and the estimates of the Sundriven disturbance \(\delta \vec {B}_\mathrm {Sun}\) show generally similar behaviour and structural features as found for Swarm Alpha, although there are also some differences. Finally, “Conclusions” section summarises the findings and provides perspectives regarding further improvements of the method.
Characterisation and calibration with scalar residuals

\(\beta = +90^\circ\): Sun directly from \(y\) (i.e. from the left during nominal flight)

\(\beta = 90^\circ\): Sun directly from \(+y\) (i.e. from the right)

\(\beta = 0^\circ\), \(\alpha = 0^\circ\): Sun directly from \(+x\) (i.e. from the front)

\(\beta = 0^\circ\), \(\alpha = +90^\circ\): Sun directly from \(z\) (above)

\(\beta = 0^\circ\), \(\alpha = +180^\circ\): Sun directly from \(x\) (i.e. from the back—slightly above the boom)
Although the observed scalar residuals clearly vary with the Sun incidence angles \(\alpha\) and \(\beta\) (see Fig. 2), there is no direct mapping of \(\Delta F\) in terms of these parameters. This is a consequence of the scalar residuals \(\Delta F \approx \delta \vec {B}_\mathrm {Sun} \cdot \vec {b}_0\) being the projection of the magnetic disturbance vector \(\delta \vec {B}_\mathrm {Sun}\), onto the unit vector \(\vec {b}_0\) of the ambient magnetic field direction (Earth’s main field). The former is oriented relative to the spacecraft, while the latter is oriented relative to Earth, which results in the variations with the spacecraft local time (captured by \(\beta\)) as seen in Fig. 2. The spacecraft local time changes by 12 hours (corresponding to a change in \(\beta\) by \(180^\circ\)) within approximately \(4\frac{1}{2}\) months.
To account for the projection on to the ambient field, we consider a vector magnetic disturbance \(\delta \vec {B}_\mathrm {Sun}(\alpha , \beta )\), with each component depending individually on the Sun incidence angles. Mathematically, we describe each component of the disturbance field vector by a spherical harmonic expansion in \(\alpha\) and \(\beta\) i.e. we consider three independent spherical harmonic expansions in all.
This model characterising the Sundriven disturbance is coestimated together with a model of the temporal evolution of the VFM sensitivity and an adjustment of the preflight estimated nonorthogonality angles of the VFM sensor. For this we perform a scalar calibration via a least squares fit, minimising the discrepancy (\(\Delta F\)) between the fully calibrated and corrected measurements from the ASM and the modulus of the vector measurements from the VFM after our model has been applied. Huber weights are used iteratively to eliminate the effect of anomalous measurements (“outliers”) on the estimated models.
Model parameterisation
Model parameters
Description  Parameters  Dimension 

\(\delta \vec {B}_\mathrm{Sun}\)  \(\vec {u}, \vec {v}\)  2028 
Sensitivity, time dependent  \(s^\mathrm{B spline}\)  9 
Sensitivity, \(\beta\) dependency  \(\vec {s}_\beta\)  3 
Sensitivity, sensor temperature dependency  \(\vec {s}_\mathrm{Tsensor}\)  3 
Nonorthogonalities  \(u_1, u_2, u_3\)  3 
Total  2046 
Estimation of model parameters: inversion and regularisation
In order to estimate the 2046 model parameters from the scalar residuals we need to solve a nonlinear inverse problem. The nonlinearity arises from the treatment of nonorthogonalities (Olsen 2003).
A regularisation matrix \(\underline{\underline{\mathbf {R}}}\) is also included to help stabilise the inversion. This is necessary because the Swarm satellites operate in a tightly controlled attitude orientation which leads to a poor excitation of the VFM instrument along the axis perpendicular to the orbit plane (the east–west direction corresponding to the yaxis of the VFM sensor). Consequently, the parameters related to the yaxis are poorly determined in a scalar calibration. The regularisation matrix \(\underline{\underline{\mathbf {R}}}\) is therefore defined so that it acts on the parameters \(s_{2,\mathrm {Tsensor}}\), \(s_{2,\beta }\), \(u_1\), and \(u_3\) to force \(s_{2,\mathrm {Tsensor}} \simeq \left( s_{1,\mathrm {Tsensor}} + s_{3,\mathrm {Tsensor}}\right) /2\) (to reflect the physical properties of the VFM sensor) and also to minimise the norms \(s_{2,\beta }^2\) and \(u_1^2 + u_3^2\). \(\lambda\) is chosen to be sufficiently large to effectively impose the regularisation on the estimated model. Note that no regularisation is directly imposed on \(\delta \vec {B}_\mathrm {Sun}\) but use of truncated SVD during the inversion automatically acts to suppresses structure in regions that are not well constrained by the input data.
The starting model for the inversions is “unity”, i.e. \(\underline{\underline{P}} = \underline{\underline{S}} =\underline{\underline{I}}\), where \(\underline{\underline{I}}\) is the identity matrix, and \(\vec {u}_n^m = \vec {v}_n^m = \vec {0}\). The inversions typically converge within 25 iterations.
Results of model estimation for Swarm Alpha
Scalar residual statistics, uncorrected, and corrected data
Satellite  Weighted rms (pT)  

Uncorrected  Corrected  
Alpha  962.6  168.3 
Bravo  710.3  164.2 
Charlie  
\(F_\text {ASM}\)  632.1  172.3 
\(F_\text {AC,map}\)  862.1  527.7 
Estimated values for selected model parameters for all three Swarm satellites
Sat  Sensitivity/sensor temperature, \(s_{\mathrm{Tsensor}}\) (\(10^{6}/{^\circ }\hbox {C}\))  Sensitivity/\(\beta\) angle, \(s_{\beta }\) (\(10^{6}/{\deg }\))  Nonorthogonalities, \(u_{1,2,3}\) (arcseconds)  

Preflight  Adjustment  Preflight  Adjustment  Preflight  Adjustment  
Alpha  28.5  0.616  –  −0.125  102.386  −0.601 
28.8  0.780  –  0  217.403  −3.960  
28.3  0.945  –  0.012  −179.318  0.149  
Bravo  28.3  1.168  –  −0.132  350.880  −0.558 
29.0  1.385  –  −0.003  62.432  −2.453  
28.8  1.602  –  −0.198  −147.060  1.608  
Charlie  27.7  1.521  –  −0.090  139.140  0.094 
29.1  1.300  –  −0.038  −248.890  1.042  
28.4  1.076  –  −0.167  −109.960  0.805 
Table 3 lists the estimated \(s_{\mathrm{Tsensor}}\) and \(s_\beta\) parameters and the nonorthogonality values for all three Swarm satellites together with their estimated preflight values for the VFM instrument itself for reference. I.e. the table shows the adjustments applied in order to reduce the scalar residuals to the level indicated above.
Weighted rms values for various models, Swarm Alpha
Model  Weighted rms (pT)  Residual power (normalised) (%) 

Full model  168.3  100 
No \(s_\beta\)  176.1  107 
No \(s_{\mathrm{Tsensor}}\)  181.7  116 
No nonorthogonalities  250.2  221 
No \(\delta \vec {B}_\mathrm {Sun}\)  962.6  3269 
Conclusions
We have established a predominantly empirical model for the calibration and correction of the magnetic vector field measurements of the three Swarm spacecraft. The model is based on detailed studies of the observed scalar residuals between the measurements of the absolute scalar magnetometer, ASM, and the modulus of the measurements of the vector field magnetometer, VFM. The model has proven to be quite robust as more data are incorporated into the estimation of the model parameters, although the ambiguity of determining vector disturbances from a pure scalar calibration affects the estimated correction vectors; these corrections do change slightly (by a few tenths of a nT) as more data are added.
The estimated models reduce the scalar differences between the Swarm magnetometers to generally below 0.5 nT with rms values well below 200 pT for all three satellites and have been in operational use since April 2015 to produce corrected Swarm Level 1b magnetic field vector data (as of version 0401).
Future evolutions of the model presented here are foreseen to include changing the model of the temporal evolution of the VFM sensitivity from Bsplines to an exponentially decaying function. Analysis of \(\delta \vec {B}_\mathrm {Sun}\) also indicates that this vector is generally confined to a few, distinct directions which may be incorporated in future models. Finally, it may be possible to model the effect of the thermal capacitance using appropriate temporal filter functions which would lead to a significant reduction of the number of parameters of the model.
Data availability
The estimated disturbance vectors, \(\delta \vec {B}_\mathrm {Sun}\), are included in the operational Level 1b magnetic Swarm data products as dB_Sun.
Uncorrected data are available at ftp://swarmdiss.eo.esa.int/Advanced/ (login required, access can be requested via https://earth.esa.int/Swarm).
Declarations
Authors’ contributions
LTC carried out the inflight scalar calibration and characterisation, analysed the results, and led the writing of this manuscript. VL proposed the model for the Suninduced vector disturbance, \(\delta \vec {B}_\mathrm {Sun}\), and made the first estimations using this model. NiO and CF supported the entire project with many discussions, suggestions, and source code. All authors read and approved the final manuscript.
Acknowledgements
We would like to thank ESA for establishing and providing support to the ASMVFM Task Force with the aim of investigating the source of the scalar residuals observed in the Swarm magnetic measurements and developing a correction scheme. We would also like to thank this Task Force for its work in characterising the behaviour of the magnetic disturbance and for many fruitful discussions and inputs for this work. In particular, we would like to thank Peter Brauer from the VFM instrument team for detailed discussions on the modelling and on the characteristics of the VFM instruments. Two anonymous reviewers are thanked for their comments that helped to improve the clarity of the manuscript. This paper is the IPGP contribution 3761.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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