In situ measurement of the magnetic field using spaceborne instruments requires a magnetically clean platform and/or a very long boom for accommodating magnetometer sensors at a large distance from the spacecraft body. This significantly drives up the costs and the time required to build a spacecraft. Here we present an alternative sensor configuration and a technique allowing for removal of the spacecraft-generated AC disturbances from the magnetic field measurements, thus lessening the need for a magnetic cleanliness programme and allowing for shorter boom length. The final expression of the corrected data takes the form of a linear combination of the measurements from all sensors, allowing for simple onboard software implementation. The proposed technique is applied to the Service Oriented Spacecraft Magnetometer (SOSMAG) on board the Korean geostationary satellite GeoKompsat-2A (GK2A). In contrast to other missions where multi-sensor measurements were used to clean the data on the ground, the SOSMAG instrument performs the cleaning on board and transmits the corrected data in real time, as needed by space weather applications. The successful elimination of the AC disturbances originating from several sources validates the proposed cleaning technique.

Since very early in space exploration it has been clear that the main limitation in performing accurate magnetic field measurements comes not from the instruments themselves but rather from the strong artificial magnetic fields generated by the spacecraft carrying them. It was recognized that there are three possible approaches to mitigating this problem: one could limit the electromagnetic emissions coming from the spacecraft by going through a rigorous magnetic cleaning procedure. This is a costly and complicated engineering task and introduces limitations on building and operating other onboard instruments; see, for example,

One of the first studies on using multi-sensor measurements to clean magnetic field data measured on board spacecraft came from

Similarly to the

The proposed method is applied to the Service Oriented Spacecraft Magnetometer (SOSMAG) instrument

The remainder of the paper is organized as follows:
in Sect.

This section gives the analytical expressions for disturbances when the exact locations of the magnetic field sources and of the sensors are known. While in most cases the direct application of these expressions is not practical, this section outlines the general principle used by gradiometer-based disturbance cleaning methods, namely the possibility to express the spacecraft-generated disturbances in terms of differences between measurements taken at distinct places. The relations derived here constitute the basis of the PiCoG technique detailed in Sect.

The magnetic field produced at the position

Assuming that the ambient magnetic field is generated by distant sources and thus it is the same at the positions

Note that the

Similar relations can be written for a time-dependent quadrupole defined by its moment

For our purposes, however, it is important that

The contributions from more than one simultaneously active, arbitrary placed source with arbitrary time dependence cannot be separated from the ambient field in a simple way. However, if multiple sensors are arranged in a suitable configuration and if specific properties of the disturbers, such as known polarization or time dependence, are used, it is possible to remove disturbances generated by multiple sources.

Two magnetometers represent the minimal configuration needed to eliminate stray spacecraft magnetic fields. Many spacecraft carry two magnetometers attached at different positions along one boom. If the boom is long enough such that the distances between the disturbance sources are much smaller compared to the distances to the measurement points and if the disturbances all have either pure dipole or quadrupole character, then their

In contrast to the minimum two-magnetometer configuration, one can imagine a configuration such that for each disturber there is a sensor placed much closer to it than to all other disturbers, plus an additional sensor far away from all disturbers. Then for each sensor the far disturbers can be assimilated to the ambient field and the problem becomes the single-disturber problem discussed at the beginning of this section. Each contribution can then be separated from the ambient field independently. Such a sensor configuration is ideal and can be attained with a number of sensors placed within the spacecraft plus one sensor placed on a short boom.

If the disturbing magnetic field has a time-dependent magnitude but does not change its direction, i.e. if its variation is linearly polarized, then up to three independent, simultaneously active disturbances with mutually orthogonal variance directions can be separated using two sensors. This is done by projecting Eq. (

The SOSMAG configuration on board the GK2A spacecraft lies somewhere in between the ideal configuration above and the minimum two-magnetometer configuration. It consists of two high-accuracy magnetometers placed on a relatively short boom and a number of resource-saving magnetometers placed inside the spacecraft. As we will show in Sect.

The PiCoG cleaning technique is based on the fact that, while the ambient magnetic field does not change over the spacecraft scale, the magnitude of a spacecraft-generated disturbance in the magnetic field decreases with the distance to the disturbance source. Therefore, the disturbance can be detected – and subsequently removed from the useful signal – by comparing measurements from sensors placed at different distances to the disturbance source as outlined in Sect.

If the precise positions of the disturbers and of the sensors are known, then the transformation matrices

The magnetic field measured by the sensor

We can eliminate the ambient field by subtracting the measurements from two sensors placed at distinct positions:

If we neglect the sensor-specific disturbances, for single disturbers, the correction to be applied to the measurements consists of a linear combination of the components of the difference

We now assume that one of the terms in Eq. (

Equations (

For many spacecraft, including GK2A, many artificial disturbances are produced by simple fixed-geometry currents without phase delays, and thus their magnetic moments are fixed in direction with only their modules changing in time
(

To find the direction of the disturbance at the sensor positions, we need to assume that the variance due to the disturbance at the sensor positions determines the maximum-variance direction of the measured magnetic field. This holds either when the variance of the disturbance is much larger than the variance of the ambient field or when the variance of the ambient field does not have a preferred direction. In this case, the direction of the disturbance at both sensors can be estimated through variance analysis

Our strategy is first to isolate the disturbance as the maximum-variance component of the differences

The components of the magnetic field at the sensor

The superscript “1” in Eq. (13) stands for the first-order correction. Note that, while the left-hand sides and the first term of the right-hand sides of Eq. (13) are represented in the VPS of the measurements at the sensor

Since the difference

If

In matrix form the above relation can be written as

While not required, the special case in which the disturbance source is collinear with the two sensors is instructive. In this case, the direction of a linearly polarized disturbance will be the same at both sensors, therefore the same coordinate system will be used for Eq. (13). Substituting

Since the corrected magnetic field should be independent of the disturbing magnetic field

For

Comparing the above with Eq. (

Further corrections can be iteratively applied as long as the stray fields from different disturbers do not have the same direction at the magnetometer location. The iteration relation from order

Using Eqs. (

The corrected field

Ideally, the hardware should consist of a “main”, least disturbed sensor and additional sensors close to each major disturbance source as described in Sect.

The components in the OB sensor system of the uncorrected measurements taken by the four magnetometers onboard GK2A on 4 March 2019.

The GK2A spacecraft launched on 4 December 2018 in a

The placement of the sensors can be seen in Fig. 6 of

As far as magnetic cleanliness is concerned, GK2A is a black box; i.e. no access to spacecraft operation time tables and to satellite-specific housekeeping data is available to aid the cleaning of the magnetic field data. Therefore the cleaning process must be based exclusively on the magnetic field measurements. Our goal is to eliminate the time-dependent spacecraft-generated disturbances from the FGM measurements. The strategy we adopt in order to take maximum advantage of the high accuracy of the FGMs and of the placement of the AMRs close to the disturbance sources is to first use the AMR measurements to clean the data from both FGMs and then use these corrected measurements to clean each other.

When a disturbance is much stronger at one sensor – as is the case for the AMR sensors – the scaling factor

Figure

For the sake of clarity, in the following we use the index

We select the time interval [15:10, 16:15] to isolate the targeted disturbance and use it to determine the variance directions of the disturbance and the scaling factors which give us the correction matrices. To lift the indetermination of the sign of the scaling factor

The angle between the direction of the disturbance at the AMR1 sensor and the direction of the disturbance at the inboard FGM sensor is

The difference

The higher-order corrections should identify and eliminate disturbances roughly ordered by their strength at the AMR1 location. However, attempting the second-order correction only introduces spurious data in the FGMs measurements, increasing their variance. This is because the noise level of the AMR sensors is higher than the noise level of the FGM sensors and the AMR1 noise is added to the corrected measurements according to Eq. (

Since the data cleaning on board the spacecraft should not require frequent updates of the correction parameters once uploaded to the spacecraft, it is necessary that the determined

The cleaning parameters resulting from the sliding window scan for the first-order correction of the outboard FGM. The top panel shows the number density of the maximum-variance direction

We now use the AMR1-corrected FGMO and FGMI measurements given by Eq. (27) as a starting point in the iteration Eq. (

Unlike the single-step disturbance we dealt with in Sect.

In order to increase the precision of the cleaning and to have an indication as to the stability of the determined parameters, we compute the cleaning parameters using sliding windows covering the entire 24 h interval. For each window

The initial AMR1-corrected FGM inboard measurements represented in the inboard VPS are plotted with the black lines. The first-order correction is plotted with red. Mean values were subtracted.

The disturbances can be much better identified in the difference

The spike-like disturbances appear now in the

The much smaller amplitude of the higher-frequency disturbance in the

For the first-order correction we target the highest-frequency disturbance by choosing the same window length of 100 s used to compute the VPS for the difference plotted in Fig.

Since the disturbances are larger at the inboard sensor, the effect of the correction is better illustrated for it than for the outboard sensor. The first-order correction of the inboard measurements for the first 4 h of the day is plotted in Fig.

The first- (black) and the second-order correction (red) for the inboard FGM sensor. Mean values were subtracted.

The magnitude of the spike-like disturbance is much reduced in the

For the second-order correction we target the remaining spike-like disturbance by choosing a window width of 700 s. Figure

The step-like disturbance and traces of the spike-like disturbance still remain in the

The second- (black) and the third-order correction (red) for the inboard FGM sensor. Mean values were subtracted.

We made use of the different characteristic timescales of the three disturbances treated in this section to help decouple them from one another even if their maximum-variance directions were not orthogonal and even if the amplitudes of the spike-like disturbances were not much different from the amplitudes of the step-like disturbances. If the disturbances had had the same timescales, these non-ideal conditions would have prevented the PiCoG cleaning method from working, unless some other specific properties of the disturbances could have been used to help decouple them.

To check the stability of the cleaning parameters, we determine them for every Sunday in 2019 with available data. The procedure produces very similar results apart from three instances when the ambient magnetic field was very disturbed. After eliminating the three outliers, we computed the standard deviations for the principal-component directions and for the scale factors, displayed in Table

The standard deviations for the first two orders are very small, indicating very stable cleaning parameters for the high-frequency disturbance and for the spike disturbance. The third order, used to clean the step-like disturbance, shows larger deviations, especially for the outboard maximum-variance direction. This is because of the small contribution of the step-like disturbance at the outboard sensor which makes the procedure susceptible to the influence of the ambient magnetic field.

Standard deviations for the directions and scale factors of the correction of FGMI data using FGMO measurements, together with the corresponding maximum deviation of the corrected magnetic field.

The final combined correction result for 4 March 2019 in the sensor system. The black lines show the original measurements taken by the outboard FGM; the red lines show the corrected data. The DC offset was restored to the value before the correction.

Since the onboard correction is designed as a one-step linear combination of the measurements from different sensors, it cannot follow the iterative procedure described in Sect.

Here

The AC correction described in Sect.

In practice, there are additional DC offsets affecting the measurements, which are treated in a separate cleaning step. The vector

It follows from Eq. (29) that the sum of the

Applying Eq. (

The

Even though we were able to eliminate most of the magnetic field disturbances on board the GK2A spacecraft, we need to be aware of the limitations the proposed method is subject to. We have already seen that, due to other disturbances or due to the ambient magnetic field variations, the maximum-variance direction might not coincide with the polarization direction of the disturbance to be removed. This difference will cause non-zero projections of the disturbance on the intermediate- and minimum-variance direction components which are not removed by the current applied correction. They will be reduced, however, by the next correction if the targeted sources lie close to each other. A disturbed ambient magnetic field may also interfere with the determination of the scaling factors. While there are ways to mitigate these effects, they are not within the scope of the present work.

The error due to sensor-specific disturbances and to the quadrupole disturbance introduced by PiCoG dipole disturbance correction. The

An important benefit of the PiCoG method is the ability to treat up to three separate disturbance sources using measurements from two sensors. In order to be able to decouple the individual disturber contributions, two conditions must be satisfied: the disturbances must have well-defined polarization directions, and these directions must be orthogonal to each other. This may seem a strong condition to impose. However, apart from moving mechanisms such as reaction wheels, many, if not most, of the magnetic disturbances from a spacecraft come from current loops without phase delays and are therefore linearly polarized. The orthogonality, on the other hand, is not guaranteed. Even in the non-orthogonal case, disturbances coming from sources close to each other compared to the distance to the sensors share the same scaling factor (if both are either dipoles or quadrupoles) and are therefore removed together. A possible way to treat non-orthogonal disturbances coming from positions separated by large distances compared to the distances to the sensors is first transforming the data to a non-orthogonal system with its axes aligned with the maximum-variance directions of the three largest disturbers. This exercise is left for future examination.

For each correction order, the disturbance to be removed has to be decoupled from the other disturbances. This is the case if the targeted disturbance amplitude is much larger than the amplitudes of the other disturbances, as assumed in Sect.

One important class of error sources is additional disturbances which do not follow the determined scaling factor

To estimate the error introduced by the sensor-specific noise combined with a quadrupole contribution additional to a dipole disturbance to be removed, let us assume a simple collinear geometry: a disturber placed in the origin of the coordinate system producing a disturbance characterized by both a dipole moment

The corrected field is obtained by applying Eq. (

The initial magnitudes of the disturbances at all sensors and the final magnitudes in the corrected data for 04/03/2019. For the MD and for the spikes the sign shows the direction of the disturbance. AMR1 does not detect a quiet interval; therefore we cannot estimate the high-frequency disturbance magnitude at AMR1. The MD and the step magnitudes are defined as the size of their ramps. The magnitudes of the spikes are equal to the spike heights/depths. The high-frequency-disturbance magnitude is defined as the peak-to-peak amplitude. Samples of the disturbances affecting the

Similar to the findings of

A way to estimate the overall performance of the cleaning is to compare the power spectral densities (PSDs) of the initial measurements with the power spectral densities of the cleaned data as shown in Fig.

The success of the cleaning procedure can also be estimated for each individual disturbance class. The initial magnitudes of the disturbances targeted for cleaning are shown in Table

For the midnight disturbance we separated the leading ramp occurring around 15:00 UTC from the abrupt trailing ramp about 1 h later. The magnitude is computed as the difference between the median over 1.5 min of the field before and after the ramp. The leading ramp is reduced from about 34 nT in the FGMO measurements to less than 2 nT in the corrected measurements. The trailing ramp is reduced from 40 nT to about 1 nT. For the components, a positive sign denotes an upward ramp, and a negative sign a downward ramp.

The ramps of the step-like disturbances are symmetric; therefore we do not differentiate between the leading and the trailing ramps. The magnitudes are computed in the same way as for the MD. The mean step magnitude is reduced from 2.5 to 1.3 nT. However, note that the

The magnitude for the spikes was computed as the difference between the value of the peak of the spike and the median over 20 s intervals 5 s before and 5 s after the peak of the spike. For 4 March 2019 we obtain a mean magnitude of 13.8 nT for the initial FGMI measurements, 4.9 nT for the initial FGMO measurements, and 0.3 nT for the corrected measurements. For the components, a positive sign denotes upward spikes, and a negative sign downward spikes.

To estimate the reduction of the high-frequency disturbance, we use as disturbance-free etalon the quiet 10 min interval visible in Fig.

We propose a multi-sensor method for removing spacecraft-generated AC disturbances from magnetic field data. The method employs principal-component analysis to decouple multiple disturbance sources and minimize the introduction of artefacts to the components free of the targeted disturbance.

A pair of sensors can resolve up to three independent disturbers. While no prior knowledge on the disturber source is required, linear polarization of the disturbance is assumed, and the polarization direction of different disturbers should ideally be mutually orthogonal. The method is robust enough to provide sensible results even if these assumptions are not strictly met. Of course, specific situations may provide additional opportunities to help separate distinct disturbers. One example is using the different characteristic timescales of the disturbances to determine the window lengths in Sect.

There are, however, situations, such as non-orthogonal disturbances from sources with large spatial separation compared with the distance to the sensors, in which two sensors are not enough to remove the disturbances with the described method. Non-linearly polarized disturbances, as those produced by reaction wheels, need special treatment not covered by this work.

We applied the PiCoG cleaning method to the GK2A SOSMAG sensor configuration by first using the spacecraft-body-mounted AMR sensor measurements to remove large disturbances from the two boom-mounted FGM sensors. Three distinct types of disturbances were then removed using the two FGM sensor measurements: high-frequency disturbance in the range of less than 1 min, spikes occurring every 10 min, and steps occurring at intervals above 1 h.

We proved that on a specific day the method was able to reduce the spectral power of magnetic field disturbances by at least a factor of 7.8 in the period range of 2 s to 1 min and by 3.9 in the period range of 1 min to 6 h. These values are representative for the performance of the method over the entire year of 2019.

The final correction takes the form of a linear combination of the different sensor readings whose coefficients were determined on the ground. These coefficients were uploaded to the GK2A spacecraft, allowing for in-flight removal of spacecraft disturbances and near-real-time delivery of cleaned magnetic field data, essential for space weather applications. In the future we shall apply the PiCoG method for post-processing of data from other spacecraft, for example from BepiColombo

SOSMAG data can be requested from the European Space Agency (ESA) and from the National Meteorological Satellite Center (NMSC) of the Korea Meteorological Administration (KMA).

ODC developed the PiCoG algorithm and implemented it numerically. HUA and WM suggested the gradiometer approach for AC cleaning and helped in refining the algorithm. MD contributed with data processing and helped in refining the algorithm. OH contributed with technical expertise about the FGM sensors and electronics. FP contributed with theoretical expertise in magnetometer calibration. HUA, MD, and WM contributed to the theoretical development and technical implementation of the SOSMAG concept on GK2A.

The authors declare that they have no conflict of interest.

We acknowledge support by the German Research Foundation and the Open Access Publication Funds of the Technische Universität Braunschweig. We are grateful to the editor, Lev Eppelbaum, and to two anonymous reviewers for their comments and suggestions, which helped us improve this work.

This research has been supported by the DLR (grant nos. 50OC1803 and 50OC1403) and the ESA (grant nos. 4000105630 and 4000117456).This open-access publication was funded by Technische Universität Braunschweig.

This paper was edited by Lev Eppelbaum and reviewed by two anonymous referees.