To clarify the geometry of the problem, we follow the formalism introduced by , which builds on . We adopt their notation for consistency, while noting that the final implementation of our coupling matrix C differs slightly in form while yielding identical results. The conversion matrix Γ transforms from the non-orthogonal sensor axes coordinate system (red axes in Fig. a) into an orthogonal sensor package coordinate system (Px, Py and Pz), represented by the blue axes in Fig. . With p^j denoting the unit vector along Pj, this can be written as Eqs. (4)–(6): 4S1=sin⁡θS1p^x+cos⁡θS1p^z,5S2=cos⁡ϕS12sin⁡θS2p^x+sin⁡ϕS12sin⁡θS2p^y+cos⁡θS2p^z,6S3=p^z, and thus 7Γ=sin⁡θS10cos⁡θS1cos⁡ϕS12sin⁡θS2sin⁡ϕS12sin⁡θS2cos⁡θS2001-1. The angles θS1, θS2, and ϕS12 define the internal geometry of the sensor axes (see Fig. a). The subsequent rotation matrix Σ aligns the sensor-package system with the spacecraft spin axis which is represented in black in Fig. b: 8Σ=cos⁡σPx0-sin⁡σPx010sin⁡σPx0cos⁡σPx×1000cos⁡σPy-sin⁡σPy0sin⁡σPycos⁡σPy, where σPx and σPy describe the tilt of the spin axis relative to the sensor package coordinate system (see Fig. b). Finally, the azimuthal rotation around the spin axis by the angle ϕa (not shown in Fig. ) is represented by 9Φ=cos⁡ϕa-sin⁡ϕa0sin⁡ϕacos⁡ϕa0001. The rotation around the spin axis arises from the mounting orientation of the FGM on the spacecraft. The angle ϕa is required to transform the measurements from the sensor coordinate system into a physically meaningful spacecraft-fixed coordinate system, from which further transformations into other coordinate systems, such as geocentric systems, can be performed. Although not required for the subsequent analysis, this rotation is included here to complete the description of the coordinate transformation. In Fig.  we show the evolution of the calibration parameters over the course of the mission. Whereas Γ(θS1,θS2,ϕS12) remains nearly constant throughout the mission for a well-constructed magnetometer as can be seen in Fig. c, f, and i, Σ(σPx,σPy) can vary over time due to spacecraft maneuvers and gravity gradients that alter the moment of inertia and thereby the spin axis orientation. This is present in Fig. a and d for more details on the coupling matrix see.

The angles αS1, αS2, and αS3 (see Fig. c) can be obtained from the projection of the spin axis onto the sensor axes using the geometric definition of the scalar product of two vectors, cos⁡θ=u⋅v/(|u|⋅|v|). In the orthogonal spin axis-aligned coordinate system, the spin axis unit vector is given by z=[0,0,1]T. Using the transformation matrices, the calculation of the angles can be expressed as: 10αS1αS2αS3=cos⁡-1Γ-1⋅Σ-1⋅Φ-1⋅z=cos⁡-1sin⁡θS1sin⁡σPx+cos⁡θS1cos⁡σPxcos⁡σPycos⁡ϕS12sin⁡θS2sin⁡σPx+sin⁡ϕS12sin⁡θS2cos⁡σPxsin⁡σPy+cos⁡θS2cos⁡σPxcos⁡σPycos⁡σPxcos⁡σPy. Assuming that θS1, θS2, and ϕS12 remain constant (cf. Fig. ), the parameters σPx and σPy can be derived from Eq. () using independently known values of αS1, αS2, and αS3 (see Sect.  for details).

The gain matrix G applies the sensor-specific conversion factors into meaningful physical units: 11G=gGp000Gp/g000Ga, where g is the ratio of the spin plane gains GS1/GS2, and Gp(=GS1GS2) and Ga are the absolute gains for the spin plane and the spin axis sensors, respectively . To determine the projected spin axis contribution BpSA from the measurements of the spin plane sensor axes data BS1 and BS2, we first subtract the offset vector OS and converting to physical units by applying the corresponding gain factors. The conversion to the modified raw data Braw is calculated by: 12Braw,1=gGp(BS1-OS1),13Braw,2=Gpg(BS2-OS2). Subsequently, we apply a low-pass Butterworth filter e.g. to Braw,1 and Braw,2 in order to suppress high-frequency variations while preserving the spin modulated signal. The filter is applied twice, once forward and once backwards, using a second-order filter with a cutoff frequency of fcut=2ωs, with ωs=2π/Tspin denoting the angular spin frequency. This results in the low-pass filtered data Blp,1 and Blp,2. The resulting filter has zero phase and an effective order of four e.g.. The low-pass filtered data are then fitted independently with a least-squares fit. The corresponding fitting function is given by: 14Bfit(τ)=(A+B⋅τ)⋅cos⁡(ωs⋅τ-φ)︸Bspinplane+C+D⋅τ︸BpSA=Bspinplane(τ)+BpSA(τ). Here, τ denotes the time within the fitting window. The term (A+B⋅τ) represents the linearly time-varying amplitude of the sinusoidal component of the fit, φ is the phase shift due to the projection of the magnetic field onto the spin plane, and the term C+D⋅τ is the residual, time-varying offset. Accordingly, the first term (A+B⋅τ)⋅cos⁡(ωs⋅(τ-φ)) corresponds to the measurement of the spin plane magnetic field in the direction of the spin plane sensor axis (Bspinplane), whereas the second term C+D⋅τ represents the projected spin axis contribution (BpSA). The parameters A, B, C, D and φ are free fitting parameters and may differ between the two components, while Tspin, and thus ωs, is provided by the state file of THE and is the same for both fits. This approach allows us to distinguish between the spin‐modulated variation within the spin plane and the slowly varying spin axis contribution.

We utilize the low-telemetry channel (FGL) of the FGM for details on the instrument and data products, see, which provides vector magnetic field data with sampling rates between 4 and 128 Hz. Since 27 July 2015, the FGL sampling rate on THE has been fFGL = 16 Hz. This sampling rate yields several tens of samples per spin and is therefore sufficient to resolve the spin modulation required for our fitting procedure. We do not use the high-telemetry channel (FGH at 128 Hz), as it is rarely available nor the onboard spin resolution product (FGS, one vector per spin), because its resampling removes the spin modulation required for the fit.

We start with Braw,1 and Braw,2 as obtained from Eqs. () and (). We then apply the low-pass filter to each component, yielding the low-pass filtered data Blp,1 and Blp,2. Finally, we fit the spin plane sensor components of Blp using Eq. () with a sliding window of 2.5 spin periods, shifted by one spin period between consecutive fits. Accordingly, τ ranges from 0 to 2.5 Tspin. We require the number of samples per fit window to be between 90 % and 110 % of the expected number,fFGL⋅2.5⋅Tspin, in order to reject windows affected by data gaps. This results in a slightly variable number of samples per window. In contrast, using a fixed number of samples would lead to windows that do not exactly cover the intended 2.5 spin periods. A comparison of both approaches (not shown) exhibits only negligible differences in the final recovered values.

As a measure of the fit uncertainty, we compute the mean squared difference between Bfit and Blp for each window: 15fmin=1n∑j=1n(Bfit(τj)-Blp(τj))2. Here, n denotes the number of measurement points per window. The quantity fmin contains everything not explained by the fit, including remaining higher-frequency contributions, sensor noise, and low-frequency variations that are not captured by the model. While the latter is the uncertainty we aim to quantify, the other contributions also affect the fit quality and must therefore be taken into account.

Each fit window yields a time-dependent estimate BpSA(τ). We evaluate BpSA(τ) at the window center, τ=1.25⋅Tspin, to obtain one estimate BpSA(tc) per window centered at time tc. This yields a time series of the projected spin axis contribution BpSA at spin resolution (≈ 3–4 s). An example of the fitting process for two consecutive fits to Blp,1 is shown in Fig. . In Fig. a, the fit window is centered at 21:41:46 UTC on 23 July 2017; in Fig. b, the window is shifted by Tspin and centered at 21:41:50 UTC.

We calculate the projected spin axis contribution BpSA independently for both spin plane component. From these two estimates, we use the result from the sensor that is more closely aligned with the spin axis, i.e., the one axis for which |90°-αS1,S2| is larger. For this sensor, the projection of the spin axis component is more pronounced.

To suppress the influence of poor fits, we calculate a weighted moving average with a window length L of 1 min: 16Bwma(t)=∑j:|tj-t|≤L/2BpSA(tj)⋅w(tj,t)∑j:|tj-t|≤L/2w(tj,t), where the weight is defined as 17w(tj,t)=wfit(tj)⋅wt(tj,t), with the fitting weight 18wfit(tj)=1fmin(tj) and the time-distance weight 19wt(tj,t)=1-|tj-t|L/2,|tj-t|≤L2,0,otherwise. The fitting weight wfit reduces the influence of poorly fitted windows, whereas the triangular time–distance weight wt emphasizes samples near the center of the averaging interval. The weighted moving average Bwma(t) is the final estimate of the projected spin axis contribution and provides an effective bandwidth of 1/L=1/60 Hz.

Having obtained the weighted moving average of the projected spin axis contribution Bwma, we reconstruct the missing spin axis measurement Br by dividing Bwma by the factor cos⁡α. This requires accurate knowledge of the angle α between a spin plane sensor and the spin axis (see Fig. c). As discussed in Sect. , these angles could in principle be derived from the coupling matrix C (see Eq. ). However, due to the loss of the spin axis measurement, a direct in-flight calibration of C is no longer feasible. This is because the regular calibration process can no longer uniquely distinguish whether a constant offset in the raw data originates from the true spin axis signal, from remaining spin plane offsets, or from inaccuracies in the angular terms of C.

To estimate the angles αS1 and αS2, we therefore use Eq. () in combination with the fitted BpSA and a modeled magnetic field approximation for the spin axis component BSA. The angle determination is restricted to perigee intervals, where the high magnetic field strength ensures that small perturbations have only a minor relative impact on the derived angles, as any residual sensor offsets and their uncertainties become negligible in comparison. We define the perigee region by a minimum field strength of 10 000 nT.

As model field BSA we use the International Geomagnetic Reference Field (IGRF) superposed with the model field of , hereafter referred to as T89. The IGRF, which represents the internal part of the Earth's magnetic field, is a good approximation of the actual terrestrial magnetic field near perigee e.g. , while the T89 model provides a simple and widely used empirical description of the external magnetospheric field, parameterized by the geomagnetic kp index .

Using Eq. (), the comparison between BpSA and BSA yields updated estimates of αS1 and αS2. To evaluate the validity of this approach, we compared the estimated α values with those from the calibration matrix over a 3-year period from 2017 to 2020. The angles differ only minimally, with median deviations of ΔαS1 = 0.045° and ΔαS2 = 0.058°.

Since May 2024, the angles between the spin axis and the two sensor axes have been 89.82° for αS1 and 90.92° for αS2; the corresponding interquartile ranges are 89.81 to 89.84° for αS1 and 90.91 to 90.93° for αS2. The orientation of the spin axis changes over time due to gravity-gradient effects and orbital maneuvers. As a result, the effective coupling matrix also varies with time, making continuous updates of these parameters necessary. The updated estimates of αS1 and αS2 are used in a least-squares optimization of the parameters σPx and σPy, ensuring continued calibration of the coupling matrix C throughout the mission. For this purpose, we use the first two components of Eq. () together with the updated values of αS1 and αS2, while treating all calibration file entries other than σPx and σPy as fixed. Each angle estimate thus provides two constraints on the two unknown parameters, and all available estimates are combined into an overdetermined system that is solved by least squares.

As discussed in Appendix , additional efforts were made to further refine and improve the recovered spin axis signal. However, the analysis reveals that such attempts not only proved challenging but also led to inconsistent results. Therefore, we consider Br as the final data product used to replace the missing spin axis measurement. It should be noted that deviations between the original measurements and the recovered signal remain, which are analyzed and discussed in the following sections.

In Fig.  we present two representative qualitative comparisons from 2017: one interval in the magnetosphere from 24 April (Fig. a), together with a more detailed view from 07:00–08:00 UTC (Fig. c); one interval from 29 July in the magnetosheath including multiple magnetopause crossings (Fig. b) with a 1 h zoom from 13:30–14:30 UTC (Fig. d).

In all panels, the original measurement BFGS, recovered data Br, and model field BT89 are shown in black, red, and blue, respectively. To ensure a consistent comparison with Br, we also apply a weighted moving average with time-distance scaling and a window length of 1 min to BFGS. In Fig. a, the recovered data closely follow the original measurements, with minor differences arising at the beginning of the shown interval. Figure b presents data where THE traversed the magnetopause multiple times between 12:24 and 15:48 UTC. Magnetopause crossings, identified using the list from , are indicated by vertical gray lines. The recovered data also generally follow the original measurements, though larger deviations are observed during boundary crossings and within the magnetosheath, where the magnetic field changes rapidly. In particular, Fig. d shows pronounced over- and undershoots, which likely indicate limitations of the fitting approach under rapidly varying conditions. Notably, in both examples Br provides a far better representation of the observed field than the model field BT89, which remains valid only within the magnetosphere.

Following the validation with original 2017 measurements, we next evaluate the recovery after the spin axis component became unavailable in May 2024 by comparing the recovered data Br with simultaneous observations from the nearby reference spacecraft THA. Two representative intervals are shown in Fig. : magnetosheath observations prior to a magnetopause crossing from 8–9 June 2024 (Fig. a) and magnetosphere observations from 28–29 November 2024 (Fig. b). Enlarged views of these intervals are shown in Fig. c for 01:00–02:00 UTC and in Fig. d for 04:00–05:00 UTC.

In all panels, the reference measurements from THA BFGS, Br, and BT89 are displayed in black, red, and blue, respectively. For consistency with the temporal resolution of the recovered data Br, the THA reference measurements were also smoothed using a weighted moving average with time-distance scaling and a 1 min window. The recovered data are generally consistent with the reference measurements in both intervals, although enhanced deviations occur near the magnetopause and within the turbulent magnetosheath (Fig. a and c). In particular, Fig. c again highlights the larger differences in the magnetosheath. In the magnetosphere interval (Fig. b), discrepancies are visible between 02:45–03:30 and 06:00–08:30 UTC, which may reflect spatial separation between THE and THA (≈ 1 RE). Without the original spin axis measurements from THE, this cannot be verified. In both examples, within the magnetosphere the recovered data provide a substantially better representation of the observed reference field than the model BT89.

The examples suggest that the accuracy of the recovery depends on the spacecraft's plasma environment. To quantify this dependence, we assess the performance of the recovery method in different regions of near-Earth space. Specifically, we divide the data from 2017 to 2020 into the following regions: the magnetosphere, the magnetopause, the magnetosheath, and the solar wind. The solar wind is further subdivided into quasi-parallel and quasi-perpendicular regions upstream of the bow shock. Within the magnetosphere, we distinguish between low (|BFGS| < 50 nT) and high field (BFGS > 50 nT) intervals.

To identify the different regions, we use the results from who applied a combination of unsupervised machine learning methods to classify THEMIS observations into magnetosphere, magnetosheath, and solar wind observations. We make use of their publicly available dataset for THE, in which each data point is assigned to one of these regions. Magnetopause and bow shock crossings are identified by changes in the region label. In addition, to further divide the dataset, we also use magnetic field data from the FGM (fully operational in 2017 to 2020) and the omni directional ion energy flux Eflux ESA, .

Magnetospheric data are separated into low- and high-field intervals to avoid biasing the statistics, as larger absolute deviations are expected close to perigee where the magnetic field is strongest. We define the low field magnetosphere by data points that fulfill |BFGS| < 50 nT and are labeled as magnetosphere. Data points in the magnetosphere for which 50 nT < BFGS are applicable belong to the high field magnetosphere. Data within ± 15 min of each magnetopause crossing are attributed to the magnetopause region. The magnetosheath is defined by the machine learning labels, except for the points near the magnetopause crossings. The solar wind dataset is used as specified by the model and then divided into quasi-parallel and quasi-perpendicular by having Eflux at 10 keV higher and lower than 20 000 cm-2s-1sr-1, respectively. Manual inspection confirmed the robustness of these assignments, with few outliers that were removed from the subsets.

We observe 6 754 802 and 1 994 568 data points in the low and high field magnetosphere, 641 029 data points around the magnetopause, and 2 077 719 in the magnetosheath, respectively. In the solar wind, however, we observe 250 187 and 182 847 data points in the quasi-parallel and quasi-perpendicular regions, respectively. The large differences in the data quantities can be attributed to the orbital configuration of the THEMIS mission. For each region, we calculate the difference between the recovered Br and the original spin axis measurements BFGS: 20δB=Br-BFGS. As in the qualitative comparison, BFGS was additionally smoothed using time-distance scaling and a 1 min window. The distributions of δB for the different regions are shown in Fig. .

Figure a shows δB for the low-field magnetosphere. The distribution is nearly centered around zero, with a slight positive shift and a median value below 1 nT. In the high-field magnetosphere (Fig. b), the distribution is broader and also shifted to positive values, with a median value below 2 nT. Near the magnetopause (Fig. d) and in the magnetosheath (Fig. e), the distributions are comparable in shape to the high-field magnetosphere and exhibit median values, below 1 nT and below 0.5 nT, respectively. In the solar wind, both the quasi-parallel and quasi-perpendicular cases (Fig. c and f) show narrow distributions centered near zero, with median values below 0.5 nT.

The position of the median reflects the quality of the spin plane offset determination, with correctly chosen offsets resulting in median values close to zero. The width of the distributions, in contrast, reflects the quality of the fit and of the recovery method, with broader distributions indicating poorer performance. Larger deviations between the recovered and original measurements occur in the high-field magnetosphere, as reflected by the larger interquartile range (> 10 nT). This is expected since absolute differences become more pronounced at higher field strengths, where small relative calibration errors or filter effects can lead to larger absolute deviations.

The smallest absolute deviations are observed in the solar wind with interquartile ranges < 6 nT. The slightly larger deviations in the quasi-parallel case can be attributed to the enhanced turbulence and wave activity in the foreshock region, which complicate the fitting process (see Sect. ). The distributions in the magnetosheath and around magnetopause crossing, however, are markedly broader. Their interquartile ranges are comparable to those of the high-field magnetosphere (> 10 nT), even though the magnetic fields themselves are much weaker. These two broader distributions can be explained due to faster changes and higher fluctuation levels in the magnetic field, which likewise complicate the fitting process there.

A somewhat different picture emerges when considering relative deviations. Using the absolute relative deviation, |δB|/|BFGS|, we find only small relative errors in the high-field magnetosphere, with a median value of 5 %. Moderate relative deviations occur in the low-field magnetosphere and near magnetopause crossings, both with median values of about 23 %. In contrast, substantially larger median values are found in the magnetosheath and the solar wind, where they reach 80 % and 115 %, respectively. This mainly reflects the smaller magnetic field strengths in these regions, for which even small absolute residuals correspond to large relative deviations; in the magnetosheath, enhanced fluctuations further increase this effect. Consequently, when the recovered spin axis field is close to zero in the solar wind, even its sign should be interpreted with caution.

To place the differences δB in the context of the intrinsic sensor noise, we estimate a lower bound for the expected reconstruction uncertainty. Following standard random-signal theory e.g., the variance of filtered stationary noise is given by 21σ2=∫0∞|H(f)|2SB1/2(f)2df, where SB1/2(f) is the amplitude spectral density of the sensor noise and H(f) denotes the effective transfer function of the reconstruction method. The quantity σ corresponds to the root-mean-square (RMS) noise amplitude, i.e., the standard deviation of the noise. In our case, the spin axis field is reconstructed from its projection onto the spin plane sensor axes, therefore the sensor noise is amplified by the same geometric factor as the signal itself.

To obtain a simple estimate, we approximate the relevant noise level by the sensor amplitude spectral density near the spin frequency fs, SB1/2(fs)≈ 10–15 pT/Hz . We further assume that, within the relevant bandwidth, the reconstruction does not strongly attenuate the low-frequency signals that constitute the final product Br. Under this approximation, the RMS noise can be estimated by integrating over an effective bandwidth Δf, which yields 22σBr≈SB1/2(fs)|cos⁡α|Δf≈80-120pT. Due to the 1 min averaging, we approximate the effective bandwidth by Δf=1/60 Hz. The resulting values provide an estimate of the minimum residual signal deviation that can be expected due solely to sensor noise. Because the sensor noise increases towards lower frequencies, the estimate based on SB1/2(fs) should be regarded as an optimistic lower bound. Since the lower bound remains well below the interquartile ranges obtained from the comparison of Br with calibrated data BFGS, sensor noise is unlikely to be the dominant limiting factor in the recovery.