Magnetometers are key instruments on board spacecraft that probe the plasma environments of planets and other solar system bodies. The linear conversion of raw magnetometer outputs to fully calibrated magnetic field measurements requires the accurate knowledge of 12 calibration parameters: six angles, three gain factors, and three offset values. The in-flight determination of 8 of those 12 parameters is enormously supported if the spacecraft is spin-stabilized, as an incorrect choice of those parameters will lead to systematic spin harmonic disturbances in the calibrated data. We show that published equations and algorithms for the determination of the eight spin-related parameters are far from optimal, as they do not take into account the physical behavior of science-grade magnetometers and the influence of a varying spacecraft attitude on the in-flight calibration process. Here, we address these issues. Based on decade-long developments and experience in calibration activities at the Braunschweig University of Technology, we introduce advanced calibration equations, parameters, and algorithms. With their help, it is possible to decouple different effects on the calibration parameters, originating from the spacecraft or the magnetometer itself. A key point of the algorithms is the bulk determination of parameters and associated uncertainties. The lowest uncertainties are expected under parameter-specific conditions. By application to THEMIS-C (Time History of Events and Macroscale Interactions during Substorms) magnetometer measurements, we show where these conditions are fulfilled along a highly elliptical orbit around Earth.

The investigation of the plasma environment in the heliosphere, around
planets, moons, comets, or other solar system bodies, requires accurate
in situ observations of the magnetic field. Magnetometers on board spacecraft
can provide these key measurements if accurately calibrated on the ground and in
flight. The calibration process delivers the parameters needed to convert raw
magnetometer measurements into magnetic field observations

Ground calibration of magnetometers is facilitated by rotating them in
Earth's magnetic field

The other four (spin-unrelated) parameters are the absolute gains in the spin
plane and along the spin axis, the spin axis offset, and the angle of
rotation of the sensor about the spin axis. Gains and angle can be derived in
flight through comparison of magnetic field measurements with the International
Geomagnetic Reference Field (IGRF) or the Tsyganenko field models, which are
fairly accurate close to Earth

From the preceding paragraphs, the reader might get the impression that
in-flight calibration of magnetometers on spinning spacecraft is a solved
issue, and in theory this is the case. However, as we will show in the
following sections, the published methods for spin-aided calibration

This paper aims at identifying deficiencies and suggesting improvements with
respect to the calibration equations (Eqs.

Equations (

List of coordinate system notations used in this paper similar to Table 1
in

Sketch of the coordinate systems:

The individual link of the sensor axes to a spacecraft-fixed, spin-axis-aligned system is an issue here as it does not reflect the actual situation
on the spacecraft: there, the three sensor axes are typically packaged
together into one sensor system. One of the design criteria of modern
fluxgate magnetometer sensors is the temperature and long-term stability of
the sensor axis directions as defined with respect to the sensor package. The
angles between the sensor axes are usually well known from ground calibration
activities

In the next step, the orientation of that sensor package system needs to be
defined in a spacecraft-fixed spin-axis-aligned coordinate system. This
latter transformation is expected to change every time there is a maneuver of
the spacecraft, as fuel consumption will change the tensor of inertia and,
thus, the spin axis direction in any spacecraft-fixed coordinate system. The
spin axis direction can be defined in the orthogonal sensor package system
using two parameters or angles. During maneuvers, only those two
parameters or angles should change because the geometry inside the sensor
package should not be affected. A rotation matrix

Using the angles

To completely orient the sensor package (system) in the spin-axis-aligned
coordinate system, a rotation about the spin axis (rotation matrix

Altogether, we can replace the orthogonalization and reorientation
matrix

The gains should be stable parameters in the absence of temperature
variations. These variations in the gains can be determined from ground
calibration, resulting in a diagonal gain correction matrix

Altogether, we suggest using the following improved calibration equation:

To determine the influence of the calibration parameters on the spin tone
harmonic disturbances in the de-spun magnetic field measurements, we use a
similar mathematical approach to

First, we compute the temperature-corrected sensor output

From the factors pertaining to the first and second harmonic terms
of

Parameters and favorable conditions.

As can be seen, the first factor of the latter group (Eq.

The same is true for the complementary factor, on the right side of
Eq. (

Let us focus on Eq. (

The first set of factors in Eq. (

The remaining elevation angles

Based on the findings from the previous section, we propose algorithms to
determine the eight spin-related parameters in an iterative manner
(Sect.

The temperature-dependent gains

The offset vector

for

for

for

and

The entire interval of magnetic field measurements should be divided into
small (overlapping) subintervals of length

For each of the subintervals, the uncertainties

From here on, we use

Parameter estimates

As detailed in the previous section, Sect.

It should be noted that we are using here the modulus of the spin plane field
(

The equivalence of the approaches (using the modulus or a de-spun component)
brings up two questions: (i) why did we not use the modulus when calculating
the influences of the spin-related parameters in Sect.

The uncertainties for each subinterval are computed as suggested in line 3 of
Table

The offset estimates

The uncertainties for each subinterval are computed as suggested in line 4 of
Table

It should be noted that the same quantity

Certain intervals may be excluded from parameter determination, as some of
the underlying assumptions may not be met well. For instance, intervals
featuring large spacecraft and sensor temperature changes should be avoided, as
parameters may vary within such intervals. Hence, uncertainties in the
parameters may be significantly higher than what is reflected in the
uncertainty estimates stated in Table

To ascertain the accuracies that parameters may be determined with in
different regions of near-Earth space, on a highly elliptical orbit around
Earth, we apply the algorithms detailed above to two days (20 and 21 July 2007)
of THEMIS-C

From top to bottom:

The different regions that THEMIS-C went through during these two particular
days are best identified using the omnidirectional ion spectral energy flux
densities, measured by the electrostatic analyzer

As can be seen in Fig.

Subinterval lengths of 100 spin periods ensure good estimates of the power at
around (and double) the spin frequency

Calibration parameter estimates as a function of their respective
uncertainties. Threshold levels for blue and red marked estimates are the
same as in Fig.

Figure

The parameter estimates themselves (

In Fig.

In order to calculate the uncertainties of the offset and elevation angle
estimates (

The offsets directly influence the absolute accuracies of the magnetic field
measurements. Typically, uncertainties on the order of or below
0.1 nT are desired in low fields. Uncertainties meeting this
threshold are marked in blue in Fig.

The only purpose of linearizing the calibration equation in
Sect.

To obtain these uncertainties, a series of assumptions need to be made, e.g.,
in the form of Eqs. (

The orthogonalization angles are known to be relatively stable when compared
to the spin axis direction angles. Fortunately, as shown in
Sect.

It should be noted that both Eqs. (

Assuming instrument linearity, the uncertainty-based approach to determining the spin-related calibration parameters allows for a meaningful estimation of the error alongside any parameter updates. These errors can be compared to the uncertainties of the already known parameters, determined either on the ground or in flight. Therewith, it is possible to decide whether any update of the calibration parameters is necessary or advised or, instead, would just introduce unnecessary variations in the calibration parameters over time.

In addition, the availability of calibration parameter estimates associated
with low uncertainties, sufficient in number and quality, determines what is
possible in terms of cadence of parameter updates. This availability depends
on the orbit of the spacecraft (the presence in regions of certain field
conditions) and also on the spin period of the spacecraft. In general, short
spin periods (high spacecraft spin frequencies) are favorable, as they
increase the number of spins that may be taken into account in subintervals
of certain length. A larger number of spin periods reduces the influence of
natural field fluctuations at (double) the spin frequency, while short
subinterval lengths ensure the constancy of the parameters and environmental
conditions. In the given THEMIS-C example, the spin plane offsets

Finally we would like to note that the benefits of parameter decoupling
(i.e., a sensible choice of parameters when taking into account the behavior
of the magnetometer and spacecraft hardware) and of the uncertainty-based
determination of those parameters are not tied to the exact definitions of
the calibration equation (Eq.

Data from the THEMIS mission including FGM and ESA data are
publicly available from the University of California, Berkeley, and can be
obtained from

FP, DF, and WM conceived the study. KHF, HUA, and IR contributed key knowledge and experience in spacecraft magnetometer calibration. DC and YN helped with the derivation and interpretation of equations.

The authors declare that they have no conflict of interest.

We acknowledge NASA contract NAS5-02099 and Vassilis Angelopoulos for use of data from the THEMIS mission. Specifically, we acknowledge Charles W. Carlson and James P. McFadden for use of ESA data and Karl-Heinz Glassmeier, Hans-Ulrich Auster, and Wolfgang Baumjohann for the use of FGM data provided under the lead of the Technical University of Braunschweig and with financial support from the German Ministry for Economy and Technology and the German Center for Aviation and Space (DLR) under contract 50 OC 0302. Edited by: Valery Korepanov Reviewed by: two anonymous referees