Fluxgate magnetometers are widely used for in situ magnetic field measurements in the context of geophysical and solar system studies. Like in most experimental studies, magnetic field measurements using the fluxgate magnetometers are constrained by the associated uncertainties. To evaluate the performance of magnetometers, the measurement uncertainties of calibrated magnetic field data are quantitatively studied for a spinning spacecraft. The uncertainties are derived analytically by perturbing the calibration parameters and are simplified into the first-order expression including the offset errors and the coupling of calibration parameter errors with the ambient magnetic field. The error study shows how the uncertainty sources combine through the calibration process. The final error depends on (1) the magnitude of the magnetic field with respect to the offset error and (2) the angle of the magnetic field to the spacecraft spin axis. The offset uncertainties are the major factor in a low-field environment, while the angle uncertainties (rotation angle in the spin plane, sensor non-orthogonality, and sensor misalignment to the spacecraft reference directions) become more important in a high-field environment in a proportional way to the magnetic field. The error formulas serve as a useful tool in designing high-precision magnetometers in future spacecraft missions as well as in data analysis methods in geophysical and solar system science.

Fluxgate magnetometers perform measurements from
DC (direct current) to low-frequency magnetic field vectors
(typically up to

The goal of the current paper is to give an outline of systematic errors of calibrated fluxgate magnetometer data on a spinning spacecraft. The error of magnetic field data occurs due to the uncertainties of the calibration parameters. The error sources may combine with one another through the calibration process. We derive the full expression of calibration errors as well as a more practical, simplified expression by truncating at the first order of relative errors. The scope of our work is the error estimate of calibrated magnetometer data in a low-field environment. In practice, more effects need to be taken into account, including sensor nonlinearities, temperature dependence (temperature drift effect), and jumps in the data associated with the change in operational modes.

For a spin-stabilized spacecraft,
the magnetometer in-flight calibration
is performed by correcting for
offsets (including the spacecraft DC field),
gains, deviations from the ideal orthogonal coordinate system,
spacecraft spin-axis direction with respect to the sensor
reference direction, and rotation angle around the spacecraft spin axis.
For a nearly orthogonal unit-gain sensor system,
the measured magnetic field is transformed
into a de-spun coordinate system and is
expanded into a
Fourier series over the frequencies as

The magnetic field vector measured by the three sensors
(sensor output) is related to the ambient field by taking account of
spacecraft spin-axis direction, spacecraft spin phase,
sensor-axis directions,
sensitivities (or gains) of the sensors,
and offsets

The true or model ambient field is set
in the inertial (i.e., non-spinning)
orthogonal spacecraft spin-axis-aligned coordinate system
(the coord-1 system in Fig.

The model ambient field in the coord-1 system
is transformed into the spinning orthogonal spin-axis-aligned system
(the coord-2 system in Fig.

The magnetic field vector in the coord-2 system
is symbolically related to that in the coord-1 system as

The field is then transformed into the spinning, orthogonal
sensor package system (the coord-3 system in Fig.

The field is further transformed into the spinning,
non-orthogonal sensor-axis-aligned system
(the coord-4 system in Fig.

Finally, in the calibration procedure, the above transformations
are inverted to estimate the ambient field
from the sensor output. The estimated or reconstructed
field is expressed in the de-spun inertial coordinate system
(the coord-5 system in Fig.

If the calibration parameters are all known,
the reconstructed field

Note that the forward transformation is defined for the conversion of the sensor output (in the coord-4 system) into the magnetic field in the physically relevant system (the coord-1 system). In the error estimate study, the inverse transformation from the coord-1 system to the coord-4 system is more instructive in order to compare the calibrated magnetic field vector in the coord-5 system with the model ambient field in the coord-1 system.

Coordinate systems used in the magnetometer calibration error estimate.

The relation between the sensor-output magnetic field

Here, the set of transformation matrices is composed of
(1) the inverse rotation matrix around
the spin axis

The calibrated magnetic field vectors depend on
the ambient magnetic field (

gain ratio

absolute gains in the spin plane

offsets in the three sensor directions

spin-axis tilt angles

deviation of elevation angles
from

deviation of azimuthal angle from

rotation angle

Note that the orthogonality nearly holds such that
the elevation and azimuthal angles exhibit only a small
deviation from 90

Also the tilt angles are small and close to zero:

The relative gain and the two absolute gains are close to unity:

The sensor output in the de-spun coordinate system
(including the temperature dependence) is expressed
up to the second-lowest order of the spin frequency as in

The systematic error of magnetic field data is analytically
derived by perturbing the calibration equations
(Eqs.

Here, the function

The function

It is useful to introduce the following variables
to simplify the notations:

If the gains (both absolute and relative ones) are close to unity
(

We assume

Derivation of the error in the

Equation (

Again, as done in the calculation of the

The differences from

The error of spin-axis component is derived from
Eq. (

For a nearly unit gain in the axial direction (

Equation (

Nominal errors (as upper limits) of calibration parameters
are summarized in Table

The spin-plane-related calibration parameters are
assessed in detail by

Nominal errors of calibration parameters.
The five lines at the top (spin-axis angles, gain ratio,
azimuthal angle, spin-plane offsets, and elevation angles)
represent the in-flight calibration for THEMIS

The offsets in the spin plane (

The lesson from the in-flight calibration for
the THEMIS magnetometer data
indicates that an offset value of about

The offset in the spin-axis direction cannot be
determined from the spacecraft spin but needs to
be determined in different ways,
for example, using additional measurements
such as absolute magnetic field magnitude

The error of gain ratio in the spin plane is
minimized to the natural fluctuation amplitude
at the second harmonic of spin frequency
in the spin plane (denoted by

Sensor-axis non-orthogonality
includes errors of the elevation angles

The elevation angles

The azimuthal angle deviation

For smaller deviation angles of

The azimuthal angle

Angular deviation of the spin axis from
the normal direction of the sensor

The remaining angle is the rotation angle in the spin plane
The rotation angle can be determined in flight
using Earth's magnetic field model in the case of Earth-orbiting
spacecraft, and the method works better in a high-field environment.
For example, the rotation angle is determined to an
accuracy of

The individual error sources are combined
using the first-order expressions
(Eqs.

The combined errors are graphically displayed
in Fig.

Error of in-flight calibrated magnetometer data
for an error of magnetometer boom angle

Equations (

The error depends on the angle between the ambient field
and the spacecraft spin axis.
The gain errors, azimuthal angle error, and
boom misalignment are coupled to the spin-plane ambient field
in the spin-plane components (Eqs.

The residual component has the largest uncertainty
in Fig.

The same plot style as
Fig.

The graphical representation of the error estimates is
extended to an ambient field of up to 10 000 nT
and is plotted again for different values of rotation angle
(

The same plot style as
Fig.

The same plot style as
Fig.

Fluxgate magnetometers are widely used in
a wide range of spacecraft missions for the study of
Earth's and planetary magnetospheres, solar system bodies,
and the heliosphere.
The magnetometer and the associated calibration process
are necessarily accompanied by uncertainties
that arise from various error sources.
We conclude the error estimate on magnetometer
in-flight calibration as follows.

Errors appear both as absolute ones (which are the offsets)
and as relative ones (angle errors, gain errors).
First-order expressions (Eqs.

The largest uncertainty sources are (1) the spin-axis offset error and (2) the spin-plane rotation angle error. The offset error appears as the dominant error in the low-field environment. The spin-plane rotation angle error plays a major role in a high-field environment, particularly when the ambient field is aligned in the spin plane.

The uncertainties are obtained by perturbing
the calibration parameters
proposed by

It should be noted that the calibration parameters are treated as time-independent in our study. In reality, however, the calibration parameters (such as offsets and gains) depend on the temperature and can evolve along the orbit. A time-dependent picture of the calibration parameters needs an extensive in-flight calibration experience.

The errors associated with the uncertainties in
calibration parameters are studied in this paper.
In a low-field environment such as in interplanetary space
the sensor nonlinearity (which originates in
the nonlinearity of gain) is usually considered negligible.
In a low Earth orbit the situation may be different.
Modern sensors, which are often double wound and even triple wound, have excellent linearity (typically to an accuracy of about

Detailed derivative calculations in Sect. 2 are presented here.

No data sets were used in this article.

YN wrote, revised, and coordinated the work; FP, WM, DF, and DS participated in discussion and writing.

The authors declare that they have no conflict of interest.

This work was financially supported by the Austrian Space Applications Programme (ASAP) at the Austrian Research Promotion Agency under contract 865967. Yasuhito Narita also acknowledges financial support by the Japan Society for the Promotion of Science, Invitational Fellowship for Research in Japan (long term) under grant FY2019 L19527. Yasuhito Narita also thanks the research and administration staff members in the Hoshino laboratory group at the University of Tokyo for discussions, support, and organization during the fellowship program and the University of Tokyo Mejirodai International Village (MIV) for the arrangement and hospitality during the pleasant and productive stay in Tokyo.

This research has been supported by the Österreichische Forschungsförderungsgesellschaft (grant no. 865967).

This paper was edited by Valery Korepanov and reviewed by two anonymous referees.