Error estimate for fluxgate magnetometer in-flight calibration on a spinning spacecraft

Fluxgate magnetometers are widely used for in-situ magnetic field measurements in the context of geophysical and solar system studies. Like in most of experimental studies, magnetic field measurements using the fluxgate magnetometers are constrained to 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 procedure, and are simplified into the first-order expression including the offset errors and the 5 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 the ambient environment such as the magnitude of magnetic field relative to the offset error and the angle of magnetic field to the spacecraft spin axis are important factors. 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 high10 field environment in a proportional way to the magnetic field. The error formulas serve as a useful tool in designing highprecision magnetometers in future spacecraft missions as well as in data analysis methods in geophysical and solar system science.


Introduction
Fluxgate magnetometers perform measurements from DC (direct current) to low-frequency magnetic field vectors (typically 15 up to 10-100 Hz), and are widely applied to in situ spacecraft observations for space plasma, magnetospheric, and heliospheric research (Acuña, 2002). The fluxgate magnetometers can be mounted on a spinning spacecraft or three-axis stabilized one, depending on the individual mission concept. In particular, in-flight calibration benefits from the spacecraft spin, since 8 of 12 calibration parameters are determined by making use of the spacecraft spin. Detailed procedure for the in-flight calibration on a spinning spacecraft are presented by, e.g., Kepko et al. (1996) and Plaschke et al. (2019). 20 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.

Systematic error on in-flight calibration
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 of spacecraft spin frequency and its harmonics. 30 The magnetic field vector measured by the three sensors (sensor ourput) 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 (Kepko et al., 1996;Plaschke et al., 2019). The relation is constructed in the following fashion.
1. 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. 1) with the spin-plane component in the X direction (B X = B p ) and the spin-axis 35 component in the Z direction (B Z = B a ). There is no magnetic field in the rest spin-plane component, B Y = 0, because the coord-1 system spans the spacecraft spin axis (in the Z direction) and the ambient field in the X-Z plane.
2. 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. 1) with the magnetic field components B x , B y , and B z by refering to the spin axis as the z direction and rotating the spin plane around the spin axis by the spacecraft spin phase −ωt (here ω is defined as the 40 de-spinning frequency and −ω as the spin frequency; t the time) as 3. The field is then transformed into the spinning, orthogonal sensor package system (the coord-3 system in Fig. 1) by 45 further rotating around the spin axis by correcting for the magnetometer boom extension and a possible misalignment of the fluxgate sensor in the spin plane (with the rotation angle φ a ) and orienting the Pz axis in the sensor-3 direction with the spin axis tilt angles σ Px and σ Py (with respect to the Pz axis) to obtain the magnetic field components as B Px , B Py , and B Pz (here, P in the subscript stands for the sensor package).
4. The field is further transformed into the spinning, non-orthogonal sensor-axis-aligned system (the coord-4 system in 50 Fig. 1) by correcting for the elevation angles θ 1 (between the sensor-1 and the sensor-3 directions) and θ 2 (between the sensor-2 and the sensor-3 directions) and also for the azimuthal separation angle φ 12 (between the sensor-1 and sensor-2 projected onto the plane normal to the sensor-3 direction) to obtain the magnetic field components B 1 , B 2 , and B 3 in the directions of the sensor axes including the gains and the offsets.
5. Finally, in the calibration procedure, the above transformations are inverted to estimate the ambient field from the sensor 55 output. The estimated or reconstructed field is expressed the de-spun inertial coordinate system (the coord-5 system in Note that the forward transformation is defined for the conversion of the sensor output (in the coord-4 system) into the magnetic 60 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.
Here, the set of transformation matrices is composed of (1) the inverse rotation matrix around the spin axis Φ −1 by the rotation angle φ a , (2) the inverse rotation matrix Σ −1 correcting for the tilt of spacecraft spin axis to the Pz direction (transforming the coord-2 system into the coord-3 system), (3) the inverse conversion matrix Γ −1 (transforming the coord-3 system into the 70 coord-4 system) and (4) the inverse gain matrix G −1 . The sensor-output field is then corrected for the offset vector O s in the sensor-axis directions. There matrices are constructed as follows .
The calibrated magnetic field vectors depend on the ambient magnetic field (B p in the spin plane and B a along the spin axis) and the following calibration parameters: gain ratio g between the two spin-plane sensors absolute gains in the spin plane G p and that in the spin axis direction G a

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offsets in the three sensor directions, O 1 , O 2 , and O 3 spin axis tilt angles σ Px and σ Py to the angles in sensor package system (angle between the coord-2 system and the coord-3 system) elevation angles θ 1 and θ 2 with a relation to the deviation from 90 degree, δθ 1 = θ 1 − 90 • and δθ 2 = θ 2 − 90 • for the sensors 1 and 2, respectively 85 azimuthal angle φ 12 with a relation to the deviation from 90 degree, δφ 12 = φ 12 − 90 • rotation angle φ a in the spin plane Note that the orthogonality nearly holds such that the elevation and azimuthal angles exhibit only a small deviation from 90 degree, 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 100 lowest-order of the spin frequency as (Eqs. 24-26 in Plaschke et al., 2019): Here, the magnetic field vector (B X , B Y , B Z ) is represented in the coord-5 system in which B Z is the component of ambient . We also assume that the calibration parameters do not change over the time or along the spacecraft orbit.

Spin-plane primary component
The systematic error of magnetic field data is analytically derived by perturbing the calibration equations . The 125 error in the X component (spin-plane primary component) is denoted by ∆B X . The spin-plane primary component is assumed to be aligned with the ambient field direction in the spin plane after calibration. On the assumption of the constant spin frequency (ω = const.), the error ∆B X is derived by perturbing Eq. (17) as follows: Here, the function max(x, y) returns the larger value from two variables, x and y, and is defined as The function max(x, y) takes the largest amplitude from an elliptically-shaped time series signal such as x cos(ωt)+y sin(ωt).
After differential calculus (see Appendix), the expression of error ∆B X is arranged to that of calibration parameters (gains, offsets, and angles):
Equation (28) is sorted to the errors of calibration parameters as: Again, as done in the calculation of the X component, we take the leading terms (the first order terms) and obtain a simplified expression of the error of residual component as:
The differences from ∆B X (Eq. 27) are 2∆(δφ 12 ) and ∆φ a in the second term in Eq. (30). The appearance of ∆φ a means that the uncertainty of the magnetometer boom extension angle (the spin-plane rotation angle) causes a finite residual component, For a nearly unit gain in the axial direction (G a 1) and small misalignments (σ Px 1, σ Py 1), the expression of error estimate is simplified into:

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Equation (33) indicates that an error occurs in the spin-axis direction (1) when the offset ∆O 3 is present, (2) when the axial (absolute) gain G a has an uncertainty, or (3) when the spin axis angle relative to the sensor Z direction has an uncertainty (which introduces a mixing or projection of the spin-plane component by the spin-axis component). Spin-plane offset S1 or S2 ∆O S1/2 0.1 nT Elevation angle S1 or S2 ∆(δθ S1/2 ) 10 −3 rad

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The lesson from the in-flight calibration for the THEMIS magnetometer data indicates that an offset value of about 0.1 nT or better (i.e., smaller) can be reached using spacecraft spin .
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 Plaschke et al., 2014) or using plasma physical properties such as the nearly-incompressible fluctuation nature in the solar wind (Hedge-220 cock, 1975; Leinweber et al., 2008), the highly-compressible fluctuation nature in which the fluctuations are nearly aligned with the ambient field (Plaschke and Narita, 2016;Plaschke et al., 2017), or the magnetic null environment in diamagnetic cavities around comets (Goetz et al., 2016a, b). The uncertainty in the spin-axis offset can empirically be minimized to 0.2 nT when using the solar wind fluctuations (Plaschke, 2019) and the mirror-mode fluctuations (Plaschke and Narita, 2016;Frühauff et al., 2017). The accuracy of spin-axis offset determination can be improved when a larger amount of data is available. An 225 accuracy of 0.5 nT or 1.0 nT is considered as representative using the mirror-mode fluctuations (Schmid et al., 2020). It is also worth noting that the offset drift is up to 1 nT per year as lessons from Cluster (Alconcel et al., 2014) and THEMIS , which may be used as a nominal value of spin-axis offset error when the spacecraft stays in the magnetosphere and the in-situ offset determination using solar wind or mirror-mode fluctuations is not possible.

Gain error 230
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 F 2p ) relative to the spin-plane ambient field B p : The gain ratio can be determined to a reasonably accurate level using the spacecraft spin, down to an uncertainty of about 10 −4 . It is true that the gain ratio in the spin plane g is related to the sensitivity measurements during the 235 ground calibration through: where S x and S y are the sensitivity (absolute gain) of the two spin-plane sensors, but the gain ratio obtained from the in-flight calibration is sufficiently accurate (∆g 10 −4 ) in practical applications.

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The elevation angles ∆(δθ 1 ) and ∆(δθ 2 ) are the angles between the sensors S1 and S3, and that between S2 and S3, respectively. The angle uncertainties∆(δθ 1 ) and ∆(δθ 2 ) can be obtained both from the ground calibration and from the in-flight calibration. Errors of the elevation angles are about 10 −3 in the in-flight calibration .
The azimuthal angle δφ S12 can thus be obtained both from the ground calibration and from the in-flight calibration, and its 255 uncertainty can be sufficiently minimized down to about 10 −4 rad in the in-flight calibration .

Misalignment to the spacecraft reference direction
Angular deviation of the the spin axis from the normal direction of the sensor x-y plane is characterized by two angles, σ Px and σ Py . The error of misalignment angles σ Px and σ Py is estimated as the ratio of the spin-axis natural fluctuation amplitude at the spin frequency to the spin-plane ambient field, and the value of σ Px/y is empirically about 10 −4 rad . The angles σ Px and σ Py need the determination or knowledge of spacecraft spin axis, and cannot usually be evaluated during the ground calibration of the sensors.
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 265 example, the rotation angle is determined to an accuracy fo 0.5 • or better when using the magnetic field data around the perigee with a field magnitude of about 8000 nT. In-flight determination of the rotation angle is meaningful when the accuracy in the in-flight method is better than the knowledge from the boom design with ground verification. We take the case of BepiColombo Mio magnetometer because the magnetometer boom extension direction is known to be within an uncertainty of 0.5 • (which gives ∆φ a = 8.7×10 −3 rad 10 −2 rad) from the spacecraft design and ground verification. As we will see in the next section, 270 the uncertainty of rotation angle in the spin plane plays an important role in the final error estimate in a high-field environment.

Combined errors of calibrated magnetometer data
The individual error sources are combined using the first-order expressions (Eqs. 27,30,and 33) to evaluate the error of calibrated magnetometer data for the nominal parameters (Tab. 1). Here, the errors represent the upper limits of the three magnetic field data in three directions (spin-plane primary, spin-plane residual, and spin-axis components). The combined 275 errors are graphically displayed in Fig. 2 as a function of the ambient magnetic field in the spin-axis direction (0 • , data curves in black) and spin-plane direction (90 • , data curves in gray). Equations (27), (30), and (33) and Figure 2) indicate that the calibration error has two distinct domains: (1) the offset dominant domain in a low-field, up to an ambient field of about 1 nT when the field is along the spin axis (curves in black in Fig. 2), and up to 10 nT when the field is in the spin plane (curves in gray in Fig. 2), and (2) the ambient field-dependent domain in a high field (above 1 or 10 nT). In the low-field case, the offset 280 dominates the magnetometer data error and the offset value is expected in the range between 0.1 to 1 nT. In the high-field case, the error grows linearly with the ambient field, and the relative error is expected between 1 % (which comes from ∆φ a ) and 0.1 % (which comes from the absolute gain error and the elevation angle error).
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. 27 and 30). The spin 285 axis misalignment and elevation angle errors are coupled to the spin-axis field. The axial gain and the spin axis misalignment are coupled to the spin-axis and spin-plane ambient field, respectively, in the expression of spin-axis component (Eq. 33). The residual component has the largest uncertainty in Fig. 2, which comes from the uncertainty of spin-plane rotation angle ∆φ a . For the reference purpose, Figure 3 exhibits the combined error estimate for the error of azimuthal angle smaller than that for Fig. 2 by an order of magnitude, δφ a ∼ 10 −3 rad. In that case, the angle errors in the calibration parameters fall onto 290 the nearly same order (between 10 −4 rad and 10 −3 rad). The final error is then below 1 nT (up to an ambient field of 300 nT) even when the ambient field is along the spin axis.

Conclusions
Fluxgate magnetometers are widely used in a wide range of spacecraft missions for the studies of Earth's and planetary magnetospheres, solar system bodies, and heliosphere. Magnetometer and the associated calibration process are necessarily ac-295 companied by uncertainties that arise from various error sources. We conclude the error estimate on magnetometer in-flight calibration as follows.
1. Errors appear both as absolute ones (which are the offsets) and as relative ones (angle errors, gain errors). First-order expressions (Eqs. 27-33) (also graphically displayed in Figs. 2 and 3) are of practical use, and show that the offset errors dominate in a low ambient field (typically below 10 nT) while the relative errors (proportional to the ambient field) 300 dominate in a high ambient field. 2. 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 with the spin axis.
The uncertainties are obtained by perturbing the calibration procedure proposed by Plaschke et al. (2019). When simplified 305 into the first-order expression, the magnetometer data errors primarily represent the offset errors as constant and the errors of gains and angles as relative error to the ambient field. Our derivation sows how how the uncertainty sources combine through the calibration process both linearly (which is dominant) and non-linearly through coupling of calibration parameter errors (which is of only secondary importance when the errors of calibration parameters are small). The error formulas are presented with analytical expressions (Eqs. 27, 30, and 33), and are expected to serve as a useful tool in various applications,

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for example, to further minimize the final error in designing a magnetometer with a boom and verifying the error throughly in the ground calibration (particularly the spin-plane rotation angle) and to report the error of scientific studies which are based on magnetometer data.
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. Time-dependent 315 picture of the calibration parameters needs an extensive in-flight calibration experience.