# A DGRF 2020 candidate model only based on Swarm ASM experimental vector mode data improved through a dedicated post-calibration strategy

Gauthier Hulot, Louis Chauvet, Robin Deborde, Jean-Michel Leger, Thomas

Jager, Martin Fillion, Pierdavide Coïsson

► To cite this version:

Gauthier Hulot, Louis Chauvet, Robin Deborde, Jean-Michel Leger, Thomas Jager, et al. A DGRF 2020 candidate model only based on Swarm ASM experimental vector mode data improved through a dedicated post-calibration strategy. Earth Planets and Space, 2026, Section on Geomagnetism, 78, pp.24. 10.1186/s40623-025-02343-1. cea-05506194

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# A DGRF 2020 candidate model only based on Swarm ASM experimental vector mode data improved through a dedicated post-calibration strategy

Gauthier Hulot¹* ID, Louis Chauvet¹ ID, Robin Deborde¹ ID, Jean-Michel Léger² ID, Thomas Jager² ID, Martin Fillion¹ ID and Pierdavide Coïsson¹ ID

**Abstract**

The ESA Swarm satellites carry a magnetometry payload consisting of an absolute scalar magnetometer (ASM), a fluxgate vector magnetometer (VFM), and a set of star trackers (STR). The primary role of the ASM is to provide 1 Hz absolute field intensity measurements, while the VFM and STR provide the additional data needed to reconstruct the attitude of the vector field and produce the official nominal Swarm L1b magnetic data. Each ASM instrument, however, can be run in an experimental mode to simultaneously produce its own self-calibrated 1 Hz vector data. Such 1 Hz experimental vector data have been routinely produced ever since launch on Swarm Alpha and Bravo, except during one-week periods every month when the burst mode was activated in yet another experimental mode to produce 250 Hz scalar data. These 1 Hz experimental vector data have been used to produce the only Definitive Geomagnetic Reference Field (DGRF) 2020 candidate model only relying on such data. All other candidate models relied on either nominal Swarm L1b data or data from other satellites and ground observatories. In this paper, we report on the way we built our DGRF candidate model and on the post-calibration strategy that we used to identify and remediate a calibration issue found in both the ASM and VFM vector data. We show that this post-calibration improves the quality of the data and contributes to also improving our DGRF candidate model. Our final candidate model, only based on post-calibrated ASM data, turns out to be one of the DGRF 2020 candidate models closest to the final official DGRF model, a posteriori providing evidence of both the quality of the Swarm ASM experimental vector mode data and the value of our post-calibration strategy. This post-calibration strategy could be used to improve magnetic data from other past, present, or future missions.

**Keywords** Geomagnetism, Magnetic field modeling, IGRF, Swarm mission, Absolute vector magnetometers, Calibration

Full list of author information is available at the end of the article

<footnote>*Correspondence:
Gauthier Hulot
gh@ipgp.fr</footnote>
## Graphical abstract

<figure><img src="image_3.png" /></figure>

# 1 Introduction

The International Geomagnetic Reference Field (IGRF) model is a series of models which provides snapshots of the main geomagnetic field every 5 years since 1900. IGRF is widely used for multiple science and operational applications (see, e.g., Macmillan and Finlay 2011), and as the main geomagnetic field keeps on evolving, updates are regularly needed. Every 5 years, a new update is therefore released to finetune the description of the field and account for its latest evolution. IGRF also comes with a secular variation model which describes the expected average linear temporal evolution of the field over the next 5 years.

Maintaining IGRF is an international effort carried out by the science community under the auspices of the Working Group V-MOD of the International Association of Geomagnetism and Aeronomy (IAGA). When the time comes, this Working Group issues a call to the science community for so-called candidate models, collects and evaluates these many models, and finally uses them to produce and release a consensually agreed update of IGRF. The process to update IGRF to its 14th generation (IGRF-14) was initiated in March 2024. As is standard procedure, the call requested candidate models in three distinct categories, each meant to produce a different

model for IGRF-14: (1) a Definitive Geomagnetic Reference Field model describing the main geomagnetic field at epoch 2020.0 (DGRF2020), to update the previously released IGRF2020 for the same epoch; (2) an IGRF2025 model, designed to describe the main geomagnetic field at epoch 2025.0, based on all data available as close as possible to the time candidate models were to be provided (October 1, 2024); (3) a predictive secular variation model, SV-2025-2030, meant to describe the expected average linear temporal evolution of the main geomagnetic field over the coming 2025–2030 period.

Building IGRF models also offers a unique opportunity to compare models computed using different methods and/or data sets, through comparison of candidate models. This IGRF-14 update is particularly rich in this respect, as more independent data sets have become available since the previous update, such as the data set provided by the Macau Scientific Satellite 1 (MSS-1), launched on May 21, 2023 (Zhang 2023), which now comes in addition to the data sets provided by the China Seismo-Electromagnetic Satellite (CSES) mission launched on February 2, 2018 (Yang et al. 2021), and the ESA Swarm constellation launched on November 22, 2013 (Friis-Christensen et al. 2006; Olsen et al. 2013). Among all those missions, Swarm holds a particular
place. First, it has been the main source of absolute vector
satellite data used by most candidate models for the two
previous IGRF generations (Thébault et al. 2015a, 2015b;
Alken et al. 2021a, 2021b) as well as for the latest 14th
IGRF generation (Beggan et al. 2025a, 2025b). Second,
each of its satellites carries two different magnetometers
that can provide independent synchronous vector meas-
urements of the field: a vector flux magnetometer (VFM)
designed by the Technical University of Denmark (DTU)
(Merayo et al. 2008), and an absolute scalar magnetom-
eter (ASM) designed by CEA-Léti in France (Léger et al.
2009) and provided by the French CNES space agency
as a customer furnished instrument (CFI). Although the
nominal role of the ASM instruments is only to deliver
1 Hz absolute scalar data for both science and calibration
purposes (for calibrating the nominal vector data from
the VFM instruments), it can indeed further produce
independent self-calibrated 1 Hz ASM-V vector data.
This has already been taken advantage of early on during
the mission (Léger et al. 2015; Hulot et al. 2015; Fratter
et al. 2016) and within the context of the previous 12th
and 13th IGRF generations (Vigneron et al. 2015, 2021),
to build excellent alternative candidate models.

Such experimental ASM-V vector data are still regu-
larly produced on the Swarm Alpha and Bravo satellites
(and no longer on Swarm Charlie, which lost its ASM
operational ability on November 5, 2014), except during
one-week sessions every month for each satellite, when
the ASM instrument is run in an alternative experimental
burst mode. This mode delivers 250 Hz scalar data that
can be used for ionospheric extremely low frequency
(ELF) signal investigations (Jenner et al. 2024; Coïsson
et al. 2025; Emsley et al. 2025). Although this implies that
fewer ASM-V data have been produced since the release
of the previous 13th IGRF generation, this turns out
not to be an issue for building field models of the type
needed for IGRF. In this paper, indeed, we will show that
these ASM-V data can again be used to build an excel-
lent DGRF2020 candidate model for the 14th IGRF
generation.

This DGRF2020 candidate model is derived from a
parent field model covering a broad 124-month period
from April 2014 to July 2024. In principle, IGRF2025 and
SV-2025-2030 candidate models could have also been
derived in the same way. However, the rules of the call
for IGRF candidates restrict each lead institution to only
propose a maximum of one candidate model in each of
the three DGRF, IGRF, and SV categories. As had already
been the case for the previous 13th IGRF generation,
IPGP opted to rely on a different strategy for the produc-
tion of its IGRF2025 and SV-2025-2030 candidate mod-
els. For those candidate models, no ASM-V data were
used. The IGRF2025 candidate model was computed

following an approach derived from Ropp et al. (2020)
and Ropp and Lesur (2023) and the SV-2025-2030 can-
didate model was computed following the approach of
Fournier et al. (2021). Technical details about how these
IGRF2025 and SV-2025-2030 candidate models were
built are described in Lesur and Shi (2024) and Fournier
et al. (2025), respectively.

In the present paper, we thus focus on deriving a parent
model entirely based on Swarm ASM-V data, which we
next use to build our DGRF2020 candidate model. The
modeling methodology is very similar to that of Vign-
eron et al. (2021) previously used for building an analo-
gous DGRF2015 candidate model. We also use the same
approach for data modeling and data quality assessment.
This “twin modeling” approach consists in first building
twin ASM-V and VFM data sets, i.e., selecting data such
that synchronous ASM-V and nominal L1b (from the
VFM instrument) data are available in the ASM-V and
VFM data sets. These twin data sets can next be used to
build twin models with exactly the same procedure, so
that any difference between these twin models can unam-
biguously be attributed to disagreements between the
data produced by the ASM instrument in vector mode
and the VFM instrument. This approach was already suc-
cessfully used by Vigneron et al. (2021) to identify a cali-
bration issue in ASM-V data, which was next corrected
for to produce the improved ASM-V data used for build-
ing their DGRF2015 candidate model.

In this article, we use the same approach and identify
yet another subtle issue. This issue shows up through
systematic disagreements in the East (B<sub>ϕ</sub>) components
of the field predicted by the initial twin parent models.
These disagreements turn out to be proportional to the
East component of the field itself. Careful comparison of
the readings of the ASM and VFM instruments in their
respective reference frames allows us to trace this issue
to some disagreements in their Y components, with
the ASM instrument systematically producing slightly
larger values than the VFM instrument on both Alpha
and Bravo satellites. This points to calibration issues
affecting the Y components of the ASM instruments,
the VFM instruments or both. A strategy is introduced
to quantify the way both instruments are affected. This
dedicated strategy allows us to produce “post-calibrated”
ASM-V and VFM data, which we use to compute a new
set of twin parent models. Comparing these twin mod-
els shows a much-improved agreement in the fields they
predict, in all three B<sub>r</sub>, B<sub>θ</sub> and B<sub>ϕ</sub> components. The final
parent model built from the post-calibrated ASM-V data
is therefore used to extract our DGRF2020 candidate
model.

In what follows, Sect. 2 first describes the data we use
and the way the twin data sets are built. Section 3 defines
the way the parent models are being parameterized and optimized. Section 4 provides the results of the computation of a first set of twin models using the initial twin data sets, and highlights the need to consider a Y-component calibration issue. Section 5 provides a comparison of the ASM-V and VFM data in instrument's reference frames and confirms the Y-component calibration issue. Section 6 describes the post-calibration strategy introduced to correct for this calibration issue. Section 7 provides the results of the computation of our final set of twin models using the post-calibrated twin data sets, also illustrating the improvement in the agreement between the twin models. Section 8 explains how our DGRF2020 candidate model was extracted from the ASM-V final parent model and describes the way uncertainties on its Gauss coefficients were produced. Section 9 finally illustrates the extremely satisfying a posteriori performance of our DGRF2020 candidate model when compared to the official DGRF2020 finally adopted by the IAGA Working Group V-MOD. Section 10 concludes this paper with lessons learnt from this study.

# 2 Data used in the study

## 2.1 Data types and versions

Only data collected by the Alpha and Bravo satellites from the ESA Swarm mission are used: ASM-V data produced by the ASM instruments running in vector mode (see Léger et al. 2015), and synchronous VFM data produced from the VFM instruments of the mission, calibrated using absolute scalar values also provided by the ASM instruments following the lines of Tøffner-Clausen et al. (2016).

ASM-V data are processed within IPGP using L0 data version 0201, v9 version software (developed by CEA-Leti) to produce L1a data, XPh1B software to produce L1b data with UTC time stamps, as well as stray-field corrections from nominal ASMxAUX data version 0602, 0605, or 0606. The calibration procedure used to infer the Characterization and Calibration Data Base (CCDB, with optimized gains of 1.22 for Alpha, 1.27 for Bravo) is the one described in Vigneron et al. (2021). All ASM-V data used are expressed in the reference frame of the ASM instruments. Attitude information is recovered using the q_NEC_CRF quaternion, defining the rotation from the Star Tracker (STR) common reference frame (CRF) to the North East Center (NEC) frame, extracted from MAGxLR files versions 0602, 0605, or 0606, also used to recover satellite positions (radius/lat/long). Note that Euler angles defining the rotation between the ASM and STR CRF reference frames are therefore jointly computed with the models (see below). VFM data are nominal Swarm L1b data from MAGxLR files versions 0602,

0605, or 0606 data available on the ESA website by July 31, 2024.

## 2.2 Data selection

Data are selected to allow the parent models to cover 124 months, from April 2014 to July 2024. We first build the ASM-V dataset. The first data used are on 26/03/2014, and the last data are on 31/07/2024. Note that we did not use the first few months of available ASM-V data in 2013 and 2014, nor the latest (after July 2024), as this was not deemed necessary to derive a DGRF candidate model for epoch 2020.0.

Within that time window, we use the same data selection criteria (except for possible thresholds when using indices, see below) as those previously used by Hulot et al. (2015) and Vigneron et al. (2021), where details can be found. These criteria, based on geophysical conditions, geographical location, and data quality, can be summarized in the following way:

* Only nighttime data (Sun at least 10° below the horizon).

* Only data acquired on magnetically quiet conditions, with low magnetospheric Ring Current activity, based on $|dRC/dt|<2$ nT/h as defined in Olsen et al. (2014) and low geomagnetic activity, based on $Kp<2+$.

* Only scalar data for all (absolute) Quasi-Dipole (QD) latitudes above 55°, further requesting that the merging electric field at the magnetopause (see Olsen et al. 2014) is not too large, based on the $E_{m,12}$ averages over the past twelve minutes satisfying $E_{m,12}<0.8$ mV/m.

* For all other QD latitudes, only ASM-V vector data, unless the scalar residual (difference between scalar and modulus of vector) is larger than 0.3 nT, or the piezoelectric motor has been activated within 3 s of the data measurement (since this may produce artifacts, see Léger et al. 2015), in which case only the scalar data are considered.

* A final decimation is being used to avoid over-representation along tracks (amounting to separate data by about 30 s).

To allow the construction of a twin L1b VFM data set, a mild additional selection criterion is finally introduced to ensure that for each ASM data selected, a matching L1b nominal data version 0602/0606 (from the VFM instruments) is available and not an outlier. This consists in checking that the corresponding L1b data vector components (resp. scalar value) are within 500 nT (resp. 100 nT) of predictions from the CHAOS 7.18 model (Finlay et al.
<figure><img src="image_4.png" /><figcaption>**Fig. 1** Data distribution as a function of time and latitude (blue: scalar data; red: vector data, partially overlaying scalar data). Note the slightly less dense data coverage after 2020, due to ASM Burst mode sessions, when no ASM-V data are being acquired, more frequently occurring since that time (typically lasting one week per month on each Alpha and Bravo satellite)</figcaption></figure>

2020) up to Spherical Harmonic (SH) degree and order 20.

This data selection process resulted in two twin ASM-V and L1b datasets, each consisting of 612 360 scalar data and 2 406 813 × 3 vector data, distributed in time and latitude as illustrated in Fig. 1.

## 3 Parent model parameterization and optimization

The parent modelling strategy follows that of Vigneron et al. (2021), itself an evolution of that of Hulot et al. (2015) and of the CHAOS-4 model of Olsen et al., (2014), within which details of the parameterization and notations used can be found.

The model involves a grand total of 10 234 coefficients corresponding to:

Core and lithospheric sources (6990 coefficients):

* Time-varying internal field up to SH degree and order 13 (included), using order 6 B-splines with a 6 × 30 days knots separation. This involves 26 splines per Gauss coefficient and therefore 26 × 13 × (13 + 2) = 5070 coefficients.

* Static internal field between SH degree and order 14 (included) and SH degree and order 45 (included). This leads to an additional 45 × (45 + 2) − 13 × (13 + 2) = 1920 coefficients.

Magnetospheric sources (1012 coefficients) modeled as in Hulot et al. (2015) following Olsen et al. (2014), where details can be found (identical notations are used here):

* Remote magnetospheric sources (SH degree 2 zonal terms in Geocentric Solar Magnetospheric (GSM) frame): $q_n^{0,GSM}$, with n = 1,2 (2 coefficients).

* Near magnetospheric ring current (in Solar Magnetic (SM) frame): $q_2^m, s_2^m$ for the static SH degree 2 (5 coefficients); $\hat{q}_1^0, \hat{q}_1^1, \hat{s}_1^1$ for the regression factors (3 coefficients); $\Delta q_1^0$ solved in bins of 5 days (748 time segments, leading to 748 coefficients); $\Delta q_1^1$ and $\Delta s_1^1$ solved in bins of 30 days (127 time segments, leading to $2 \times 127 = 254$ coefficients).

Euler angles describing rotation between ASM and STR CRF reference frames:

* estimated every 10 days for both satellites: 374 and 370 time segments for, respectively, Alpha and Bravo, leading to $3 \times (374+370) = 2232$ coefficients.

Parent models are optimized using the same approach as in Vigneron et al. (2021) by minimizing the mismatch between data and model prediction, using iteratively reweighted least-squares with Huber weights and temporal damping as in Olsen et al. (2014), but only using two damping parameters $\lambda_2$ and $\lambda_3$:

* $\lambda_2$: $|d^2B_r/dt^2|$ constrained at beginning and end of dataset (i.e., March 2014 and July 2024) with the same damping value as CHAOS-4 ($\lambda_2 = 10(nTyr^{-2})^{-2}$).

* $\lambda_3$: $|d^3B_r/dt^3|$ integrated at the core surface and over the time coverage constrained with the same damping value as CHAOS-4 ($\lambda_3 = 0.33(nTyr^{-3})^{-2}$ for all Gauss coefficients except for $g_1^0$ where we use $\lambda_3 = 10(nTyr^{-3})^{-2}$).

As in Hulot et al. (2015) and Vigneron et al. (2021), geographical weights are introduced, proportional to $\sin(\theta)$ (where $\theta$ is the geographic colatitude), to balance the geographical sampling of data. Anisotropic magnetic errors due to attitude uncertainty is taken into account assuming an isotropic attitude error of 10 arcsecs following Holme and Bloxham (1996). A priori data error standard deviations are otherwise set to 2.2 nT for both scalar and vector data. The starting model used is a static model based on CHAOS-4 up to SH degree and order 13, to rely on a model built without any Swarm data, and slightly extrapolated to epoch 01/12/13, to get closer to the starting date of our parent model. But this choice is not critical. It only helps reducing computation time and has no significant influence on the final models. A total of eight iterations is
<table><caption>**Table 1** Residual statistics for all data used to produce the initial ASM-V and VFM twin parent models</caption><thead><tr><th rowspan="2"></th><th rowspan="2">N</th><th colspan="2">ASM-V (ASM-V data)</th><th colspan="2">VFM twin (VFM data)</th></tr><tr><th>Mean (nT)</th><th>RMS (nT)</th><th>Mean (nT)</th><th>RMS (nT)</th></tr></thead><tbody><tr><td colspan="6"><em>Alpha + Bravo</em></td></tr><tr><td>F</td><td>612360</td><td>-0.06</td><td>3.81</td><td>-0.05</td><td>3.82</td></tr><tr><td>F and B<sub>B</sub></td><td>3019173</td><td>0.03</td><td>2.50</td><td>0.01</td><td>2.48</td></tr><tr><td>B<sub>B</sub></td><td>2406813</td><td>0.05</td><td>2.10</td><td>0.02</td><td>2.07</td></tr><tr><td>B<sub>r</sub></td><td>2406813</td><td>-0.01</td><td>2.19</td><td>-0.01</td><td>1.54</td></tr><tr><td>B<sub>θ</sub></td><td>2406813</td><td>0.05</td><td>3.24</td><td>0.01</td><td>2.78</td></tr><tr><td>B<sub>φ</sub></td><td>2406813</td><td>-0.01</td><td>2.86</td><td>0.00</td><td>2.45</td></tr><tr><td colspan="6"><em>Alpha</em></td></tr><tr><td>F</td><td>302426</td><td>-0.14</td><td>3.94</td><td>-0.13</td><td>3.96</td></tr><tr><td>F and B<sub>B</sub></td><td>1475747</td><td>0.01</td><td>2.56</td><td>-0.02</td><td>2.54</td></tr><tr><td>B<sub>B</sub></td><td>1173321</td><td>0.05</td><td>2.14</td><td>0.00</td><td>2.10</td></tr><tr><td>B<sub>r</sub></td><td>1173321</td><td>0.01</td><td>2.27</td><td>0.03</td><td>1.57</td></tr><tr><td>B<sub>θ</sub></td><td>1173321</td><td>0.02</td><td>3.27</td><td>0.04</td><td>2.79</td></tr><tr><td>B<sub>φ</sub></td><td>1173321</td><td>0.00</td><td>2.90</td><td>0.00</td><td>2.46</td></tr><tr><td colspan="6"><em>Bravo</em></td></tr><tr><td>F</td><td>309934</td><td>0.02</td><td>3.68</td><td>0.04</td><td>3.69</td></tr><tr><td>F and B<sub>B</sub></td><td>1543426</td><td>0.04</td><td>2.44</td><td>0.04</td><td>2.43</td></tr><tr><td>B<sub>B</sub></td><td>1233492</td><td>0.05</td><td>2.07</td><td>0.04</td><td>2.05</td></tr><tr><td>B<sub>r</sub></td><td>1233492</td><td>-0.03</td><td>2.10</td><td>-0.05</td><td>1.51</td></tr><tr><td>B<sub>θ</sub></td><td>1233492</td><td>0.08</td><td>3.20</td><td>-0.01</td><td>2.77</td></tr><tr><td>B<sub>φ</sub></td><td>1233492</td><td>-0.02</td><td>2.82</td><td>0.00</td><td>2.45</td></tr></tbody></table>

N is the number of data used; mean and RMS are the Huber-weighted misfit mean and Root Mean Square values (in nT); F stands for scalar data, B<sub>B</sub> stands for the field component projected along the field direction (providing a measure of the misfit of the modulus of the vector data with respect to model prediction); B<sub>r</sub>, B<sub>θ</sub>, and B<sub>φ</sub> stand for the three geocentric vector field components

used, which is found satisfactory to ensure convergence to within the accuracy required.

## 4 Initial twin parent models

Using the ASM-V and twin L1b VFM data sets described in Sect. 2, we now apply the modeling methodology described in Sect. 3 to derive an initial ASM-V parent model together with its twin initial VFM parent model. Table 1 provides residual statistics for both the ASM-V and twin VFM models, for all data, Alpha data only, and Bravo data only. These Huber-weighted residual means and RMS are almost always (slightly) smaller than those Vigneron et al. (2021) found when building analogous twin parent models to produce their DGRF2015 candidate model (see their Table 1). Yet, both pairs of parent models have been computed in the same way (same methodology, same choice of all parameters). They only differ by the data versions used and the time period covered.

Changes in the data version since the study of Vigneron et al. (2021) reflect changes that have been introduced

in the way a so-called dBSun perturbation affecting the magnetometry payload is being corrected for. This perturbation is a function of the Sun incidence angles with respect to the spacecraft (see Toffner-Clausen et al. 2016) and is now known to be due to thermoelectric effects that produce tiny electric currents within the complex (and unfortunately poorly characterized before launch) electric grounding circuits of the thermal multi-layer insulation (MLI) blanket surrounding the magnetometry payload (Brauer et al. 2024). This perturbation was initially thought to only affect the VFM instrument. In the nominal VFM and ASM-V data versions used by Vigneron et al. (2021), only the nominal VFM data were therefore corrected for it. Progress over the past years, however, made it clear that this initial correction was partly incorrect and that corrections had to be split into two separate contributions, one applying to the nominal VFM data, the other to the ASM (scalar and ASM-V vector) data (Brauer et al. 2024). The data versions used in the present study acknowledge this and now include new dBSun corrections on both the nominal VFM and ASM-V data (the latter being now included as a contribution in the ASMXAUX data files that provide stray-field corrections to be applied to all ASM data).

The slight improvement in the residuals shown in Table 1 for our initial twin models, compared to those of Vigneron et al. (2021) for their twin models, is a good sign that the new dBSun corrections have indeed improved the quality of both nominal VFM and ASM-V data sets. This is all the more encouraging that the period covered by our initial twin models (April 2014 to July 2024, 124 months) is much longer than that of the twin models of Vigneron et al. (2021) (November 2013 to May 2019, 67 months), which would normally lead to a slight increase in data residuals.

Despite this improvement, differences between the predictions of the initial ASM-V and VFM twin parent models still reveal large-scale signatures very similar to what Vigneron et al. (2021) had found when comparing their twin parent models. Figure 2a shows a map of such differences computed at Earth's surface at epoch 2020.0 for the radial component B<sub>r</sub> up to SH degree and order 13. This map indeed shows a pattern very close to the pattern seen in Fig. 4b of Vigneron et al. (2021), which shows the B<sub>r</sub> component difference between predictions of their DGRF2015 candidate model and its twin model (each being extracted from their respective parent models for epoch 2015.0). In their conclusions, Vigneron et al. (2021) noted that this pattern could be due to the dBSun correction not being optimally implemented at the time. However, Fig. 2a clearly shows that despite the slight improvements introduced in the dBSun correction, just discussed above, the pattern originally seen in Fig. 4b of
<figure><img src="image_5.png" /><figcaption>**Fig. 2** Differences between the predictions of the initial ASM-V and VFM twin parent models for the $B_r$ (a), $B_\theta$ (b) and $B_\phi$ (c) components. Also shown in d, the $B_\phi$ component of the field predicted by the initial VFM twin parent model. Both models are computed up to SH degree and order 13, and all maps are plotted at Earth's surface (6371.2 km) for epoch 2020.0</figcaption></figure>

Vigneron et al. (2021) did not disappear, and even seems to have been slightly enhanced. This result prompted us to dismiss the dBSun effect as the main cause for this persistent disagreement between ASM-V and VFM twin parent models and to look for other possible causes.

Figure 2b and c shows analogous maps of differences between the predictions of the initial ASM-V and VFM twin parent models for the $B_\theta$ and $B_\phi$ components. These also reveal large-scale patterns. Most striking is the very strong similarity of the map of the difference in the $B_\phi$ component of Fig. 2c with the map of the $B_\phi$ component of the field predicted by the VFM twin parent model (Fig. 2d, virtually undistinguishable from the same map plotted using the ASM-V parent model). Comparing these two maps shows that for almost all locations, the $B_\phi$ component predicted by the ASM-V parent model is systematically slightly larger (in absolute magnitude) than the one predicted by the VFM twin parent model. The magnitude of the difference seems to be mostly proportional to the $B_\phi$ component of the full field. This pattern persists throughout the entire period covered by the models.

The above similarity between Fig. 2c and d leads us to suspect that another calibration issue still affects the

data. The reasoning is the following. As Alpha and Bravo are near-polar orbiting satellites, the $B_\phi$ component of the field is almost always aligned with the Y-components of both the ASM-V and VFM instruments in their own reference frames (except at very high latitude, where vector data are anyway not used in the modeling methodology, recall Sect. 2.2). This suggests that the Y component of the ASM instrument systematically overestimates the Y value it should measure, that the VFM instrument systematically underestimates its own Y component, or both. No similar matching pattern could be found for the $B_r$ and $B_\theta$ components in Fig. 2a and b and the $B_r$ and $B_\theta$ components of the full field. This further suggests that the above suspected calibration issue only affects the Y components of the ASM-V and/or VFM data.

**5 Comparison of the ASM-V and VFM data in the instrument's reference frames**

To further investigate the possibility of a calibration issue affecting the ASM-V and VFM data, advantage can be taken of the fact that the ASM and VFM instruments have very closely aligned reference axes, to within a difference in the naming and orientation conventions of these axes (Fig. 3). The $Z_{VFM}$ axis of the VFM instrument is
<figure><img src="image_6.png" /><figcaption>**Fig. 3** ASM and VFM instruments' reference frames on the Swarm satellites</figcaption></figure>

along the boom pointing toward its tip (away from the body of the satellite), the $X_{VFM}$ axis is orthogonal to the boom and points downwards in the plane perpendicular to the bottom face of the satellite body and including the boom axis, and the $Y_{VFM}$ axis completes the orthogonal frame. In contrast, the $X_{ASM}$ axis of the ASM points in the opposite direction to that of the $Z_{VFM}$ axis, the $Z_{ASM}$ axis points in the opposite direction to that of the $X_{VFM}$ axis, and the $Y_{ASM}$ axis again completes the orthogonal frame. As a result, to first order, the three components of the field measured by the ASM and VFM instruments in their respective reference frames are related through the simple transformations: $X_{ASM} = -Z_{VFM}$; $Y_{ASM} = -Y_{VFM}$; $Z_{ASM} = -X_{VFM}$. But because these reference frames are not exactly aligned in this way (due to the challenge of accurately aligning hardware on a satellite) and because the boom and optical bench mechanically linking the two instruments may suffer slight deformations in orbit, an alignment procedure has been implemented to recover the exact quaternion linking the two reference frames. As had been shown by Vigneron (2019, Sect. 3.3), mechanical deformations between the two reference frames are small and mainly occur on the long term as a function of the local time of the satellites. Within ten days, they only reach 4 arc-seconds. This offers the possibility of carrying day-by-day alignment procedures to accurately map the ASM-V measurements in the VFM reference frame and compare their measurements along each axis. This alignment procedure is directly done by IPGP when producing the ASM-V data.

The corresponding $q_{VFM}_ASM$ quaternions defining the rotation from the ASM frame to the VFM frame and the resulting $B_{ASMV}_{VFM}$ components of the ASM-V data in the VFM reference frame are provided with the ASM-V data. In what follows, we thus directly compare ASM-V and nominal VFM vector components within the VFM reference frame.

Figures 4 and 5 show the difference $\Delta Y_{VFM} = Y_{ASMV_{VFM}} - Y_{VFM}$ between the $Y_{ASMV_{VFM}}$ component of the field measured by the ASM instrument and the $Y_{VFM}$ component of the field measured by the VFM instrument, as a function of the $Y_{VFM}$ component. All values are expressed in the VFM frame. These values are 1-min averages to reduce the impact of noise. Figure 4 shows such comparisons for Alpha ascending half-orbits in 2020, plotting the data in 24 local time sectors. Figure 5 shows the same comparisons now for Swarm Bravo and when the satellite is on descending half-orbits. Very similar figures are obtained when considering ascending half-orbits Swarm of Bravo and descending half-orbits for Alpha. More generally, plotting the same figures for any year between 2014 and 2024 and both Alpha and Bravo leads to exactly the same type of behavior. These figures illustrate the fact that in all cases, i.e., for all LT experienced by Alpha or Bravo, and for both ascending or descending half-orbits, $\Delta Y_{VFM}$ is linearly related to $Y_{VFM}$.

For each local time sector and half-orbit, linear regressions of the form:

(See figure on next page.)

**Fig. 4** Alpha satellite differences $\Delta Y_{VFM} = Y_{ASMV_{VFM}} - Y_{VFM}$ between the $Y_{ASMV_{VFM}}$ component of the field measured by the ASM instrument and the $Y_{VFM}$ component of the field measured by the VFM instrument, as a function of the $Y_{VFM}$ component (all expressed in the VFM frame). Values are 1-min averages along orbit. Each panel is for all ascending half-orbits in 2020 for the one-hour interval of the LT at the location of data acquisition specified in the title of each panel. Best fit to Eq. (1) in a least-square sense is shown in red solid line (2σ of data distribution about the best fit shown in magenta solid lines), with corresponding equation shown as inset
Alpha satellite 2020 ascending half-orbits

<figure><img src="image_7.png" /><figcaption>**Fig. 4** (See legend on previous page.)</figcaption></figure>
Bravo satellite 2020 descending half-orbits

<figure><img src="image_8.png" /><figcaption>**Fig. 5** Same as Fig. 4, but for descending half-orbits of the Bravo satellite in 2020</figcaption></figure>
$$ \Delta Y_{VFM} = \alpha Y_{VFM} + \beta \quad (1) $$

can be computed. The resulting best-fits (in a least-square sense) are shown in each plot (with values of α and β provided in insets). Values of α are always comparable and of typically 2 × 10⁻⁴, though these may occasionally reach slightly higher or lower values. In contrast, values of β (offsets) remain close to zero, varying within at most −1 nT and +1 nT, whereas ΔYVFM itself varies between typically −5 nT and 5 nT.

That the β offsets vary as a function of the LT, and are also sensitive to the ascending/descending nature of the half-orbit (not shown), can be interpreted in terms of possible residual dBSun perturbations still not fully corrected for. As already noted, these perturbations, typically of a couple of nT, are indeed known to be a function of the Sun position relative to the spacecraft and are therefore sensitive to both the LT seen by the satellites and their ascending/descending orientation (see e.g., Tøffner-Clausen et al. 2016). These dBSun perturbations, however, are not sensitive to the magnitude of the field witnessed by the satellites and cannot account for the fairly stable values of the α slope at all LT. They are also independent of the orientation of the physical Y axis of the ASM and VFM instruments, which can be eastward or westward depending on whether the satellites are on ascending or descending half-orbits (recall Fig. 3). In contrast, a stable α slope is precisely what one would expect if a calibration issue was to indeed affect the Y axis of either or both the ASM and VFM instruments.

Figure 6 provides a summary of the way this α slope behaves as a function of LT for the Alpha and Bravo satellites, for ascending (solid lines) and descending (dashed lines) half-orbits. A color code is also used to allow comparison of these curves as a function of the year the observations were made (between 2014 and 2024). These plots first confirm the overall stability of the α slope at positive values of order 2 × 10⁻⁴. Significant deviations from the common overall behavior are mainly observed for Alpha in 2023, for which we currently do not have any simple explanation. Only slight dependence on the LT is otherwise observed as well as small different behaviors when considering ascending or descending half-orbits, mostly when considering LT between 7:00 and 15:00 and more clearly so on Bravo. The cause of this could be small systematic mechanical deformation of the boom linking the ASM reference frame to the VFM reference frame, which the day-by-day alignment procedure we use to compare the data cannot account for. These LT are indeed those when the boom experiences the strongest Sun exposure and thermal contrasts when moving from dark to Sun. Overall, however, these plots clearly confirm a consistent tendency of the Y component of the ASM

instrument to provide values exceeding those (in absolute value) of the Y component of the VFM instrument by a relative value of order α = 2 × 10⁻⁴, thus testifying for a calibration issue on this component on either or both the ASM and VFM instruments.

A similar investigation can be done to check for possible additional calibration issues on the X and Z components of the ASM and VFM instruments, by relying on exactly the same procedure. Figures 7 and 8 provide the same plots as in Fig. 6, but for the Z and X components in the VFM frame (i.e., to compare values provided by the X, resp. Z, axis of the ASM instrument with those provided by the Z, resp. X, axis of the VFM instrument, after alignment; recall Fig. 3). These figures starkly contrast with Fig. 6. Although some slight fluctuations are observed in Fig. 7, no systematic offsets are observed at any LT for any year, neither for ascending nor for descending half-orbits. This clearly shows that no similar calibration issue affects the X and Z components on either of the ASM and VFM instruments.

The very stable consistency between the values provided by the X axis of the VFM instrument (pointing downwards perpendicular to the boom, recall Fig. 3) and those provided by the Z axis of the ASM instrument is particularly impressive (Fig. 8). We interpret this as a possible confirmation that the fluctuations seen in Fig. 6 on top of the very clear offset (testifying for the Y component calibration issue, the possible cause of which will later be discussed in Sect. 10) may well be the result of a boom deformation corresponding to a rotation of the ASM reference frame with respect to the VFM reference frame about the X axis of the VFM instrument along the orbit, when the satellites get in the Sun after being in the dark. Such a deformation would always be in the same direction in the satellite reference frame for the same Sun exposure. It would not affect the alignment between the Z axis of the ASM and the X axis of the VFM and produce no signature in Fig. 8, as is the case. In contrast, it would produce a signature in both Figs. 6 and 7, with the largest signature to be found in Fig. 6, as the impact on the smaller Y component would be larger, with opposite effects depending on whether the satellite is ascending or descending within the same ambient field. This interpretation is also consistent with the fact that variations in the q_VFM_ASM quaternions defining the daily computed rotation from the ASM frame to the VFM frame are mainly rotations about the X axis of the VFM instrument, as noted by Vigneron (2019, Sect. 3.3).

## 6 Post-calibration strategy

Having confirmed that a calibration issue is affecting the Y component (and only this component) of the field in the ASM-V and/or the nominal VFM data, we now turn
ΔY α slope values for the Y component in VFM reference frame

<figure><img src="image_9.png" /><figcaption>**Fig. 6** Values of the α slope (Eq. 1) as a function of LT for the Alpha (top panel) and Bravo (bottom panel) satellites, for ascending (solid lines) and descending (dashed lines) half-orbits. A color code is used to allow comparison of each curve for the various years observations were made (central panel, between 2014 and 2024). Note that values for each LTT corresponds to data selected between T and T + 1 h LT</figcaption></figure>

to the issue of quantifying the extent to which each of
the two ASM and VFM instruments are affected. Our
goal is to next design a remediation strategy to produce
improved data and parent models.

The main limitation of the analysis carried out in
Sect. 5 is that it only provides evidence of a disagree-
ment in the calibrations of the Y components of the
two ASM and VFM instruments. It cannot tell which
of the two instruments is responsible for this disagree-
ment. To address this question, we need to rely on a self-
consistent strategy to test each of the two instruments
independently.

6.1 Data residuals analysis

Figure 9 provides the way forward. Figure 9a shows the map of ΔB<sub>ψ</sub> residuals of all ASM-V B<sub>ψ</sub> data used in building the initial ASM-V parent model with respect to the B<sub>ψ</sub> values this model predicts. Note that in this context, ASM-V B<sub>ψ</sub> data are the B<sub>ψ</sub> component inferred from the ASM-V vector data using the q_NEC_CRF quaternion from MAGxLR files versions 0602, 0605 or 0606, combined with the quaternion defining the rotation between the ASM and STR CRF reference frames jointly computed with the initial ASM-V parent model (recall Sect. 3). As previously noted in Sect. 4, however, this
ΔZ α slope values for the Z component in VFM reference frame

<figure><img src="image_10.png" /><figcaption>**Fig. 7** As Fig. 6 but for the $\Delta Z_{VFM} = Z_{ASMV\_VFM} - Z_{VFM}$ difference between the $Z_{ASMV\_VFM}$ component of the field measured by the ASM instrument and the $Z_{VFM}$ component of the field measured by the VFM instrument, both expressed in the VFM frame. The $\alpha$ slope values are now those of the linear regression $\Delta Z_{VFM} = \alpha Z_{VFM} + \beta$</figcaption></figure>

component is almost always aligned with the Y component of the ASM-V instrument at the latitudes of interest.

Figure 9a reveals which part of the signal the model rejects as being inconsistent with its parameterization. A clear large-scale pattern emerges. This pattern is reminiscent of the pattern seen in Fig. 2c when comparing the $B_{\varphi}$ component predictions of the initial ASM-V and VFM twin parent models. It also compares quite well with the pattern of the $B_{\varphi}$ component predicted by both models (Fig. 2d). Although such $B_{\varphi}$ residuals inevitably contain geophysical signals not captured by the model, such as small-scale internal signals (e.g., those associated with the notoriously large African Bangui and Eurasian Kursk

lithospheric anomalies, see e.g., Thébault et al. (2021)), or unmodelled external field signals (such as toroidal fields produced by poloidal currents at satellite altitude, see e.g., Fillion et al. (2023)), the similarity in the pattern found in Figs. 9a and 2d is a clear indication that the Y component in the ASM-V data may slightly overestimate the actual Y value of the field. Indeed, as the field is being modeled in terms of a potential field, if Y values of the field measured by the ASM instrument are too large to be compatible with the constraints provided by the values of the two other X and Z components, the model can be expected to reject the excess of Y value.
ΔX α slope values for the X component in VFM reference frame

<figure><img src="image_11.png" /><figcaption>**Fig. 8** As Fig. 6 but for the ΔX<sub>VFM</sub> = X<sub>ASMV_VFM</sub> - X<sub>VFM</sub> difference between the X<sub>ASMV_VFM</sub> component of the field measured by the ASM instrument and the X<sub>VFM</sub> component of the field measured by the VFM instrument, both expressed in the VFM frame. The α slope values are now those of the linear regression ΔX<sub>VFM</sub> = αX<sub>VFM</sub> + β</figcaption></figure>

To tentatively quantify the fractional excess of Y value
the ASM-V data appears to provide, we proceed in a
way analogous to what we previously did in Sect. 5 when
comparing ASM-V and VFM data, now comparing data
residuals, i.e., ASM-V data minus model prediction in
the ASM instrument's reference frame, assuming the
model provides a correct first-order representation of
the field to be measured. We again classify the data in 1 h
LT bins. Because we now only consider data that were
selected for building the initial parent model according to
the selection criteria of Sect. 2.2, no day time data with
LT between 9 and 15 and very few data between 7 and
9 and between 15 and 17 are involved (Fig. 10). In what

follows, we thus more specifically focus on nighttime LT
bins between 17 and 7 for which enough data are avail-
able to provide meaningful results. In addition, because
no significant differences could be found when consider-
ing ascending or descending orbits for such nighttime LT
when previously comparing the ASM-V and VFM data in
instrument's reference frames (recall Fig. 6), no distinc-
tion is now been made between ascending and descend-
ing orbits.

For each of these LT, we compute the
ΔY<sub>ASM</sub> = Y<sub>ASM</sub> − Y<sub>ASM_Model</sub> residuals of the Y<sub>ASM</sub>
ASM-V component with respect to the Y<sub>ASM_Model</sub> value
predicted by the initial ASM-V parent model in the ASM
<figure><img src="image_12.png" /><figcaption>Fig. 9 Maps of ΔB<sub>φ</sub> residuals of B<sub>φ</sub> data used in building each initial parent model with respect to the B<sub>φ</sub> values predicted by the corresponding initial model. a shows the map for the initial ASM-V model; b shows the map for the initial VFM model. Values in each 1° × 1° element of the maps are average values of all residuals within the corresponding latitude/longitude range</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10 Number of vector data selected for the production of all parent models as a function of Local Time at the location of data acquisition (blue/red shading corresponds to, respectively, Bravo/ Alpha data). Note that by construction, this distribution is the same for the ASM-V and twin VFM data sets</figcaption></figure>

reference frame. We next compute best linear fits (again in the least-square sense) between these ΔY<sub>ASM</sub> residuals and the corresponding predicted Y<sub>ASM_Model</sub> values:

$$ \Delta Y_{\text{ASM}} = \alpha_{\text{ASM}} Y_{\text{ASM\_Model}} + \beta_{\text{ASM}} \quad (2) $$

Figures S1 and S2 in the Supporting Information provide plots of ΔY<sub>ASM</sub> as a function of Y<sub>ASM_Model</sub> for each of the LT and each of the Alpha and Bravo satellites, using the same plotting convention as in Figs. 4 and 5. Figure 11a summarizes the recovered values (and

2σ uncertainties) of the α<sub>ASM</sub> slopes as a function of LT. These α<sub>ASM</sub> slopes (and the β<sub>ASM</sub> offsets, also recovered) will be discussed later.

Exactly the same analysis can be carried out independently when considering VFM data which we now compare to the initial VFM parent model. Figure 9b shows the map of ΔB<sub>φ</sub> residuals of all VFM B<sub>φ</sub> data used in building the initial VFM parent model with respect to the B<sub>φ</sub> values this model predicts. VFM B<sub>φ</sub> data are now the B<sub>φ</sub> component inferred from the VFM data using the q_NEC_CRF quaternion from MAGxLR files versions 0602, 0605 or 0606, combined with the quaternion defining the rotation between the VFM and STR CRF reference frames jointly computed with the initial VFM parent model. Contrary to Fig. 9a, b does not show an overall tendency of the ΔB<sub>φ</sub> residuals to correlate, but rather a slight tendency to anticorrelate, with the B<sub>φ</sub> component predicted by the VFM initial parent models (Fig. 2d). We now interpret this as a (faintier) indication that the Y component in the VFM data may slightly underestimate the actual Y value of the field.

As was done for the ASM-V data, we thus repeat the analysis that led to Fig. 11a, now using the ΔY<sub>VFM</sub> = Y<sub>VFM</sub> - Y<sub>VFM_Model</sub> residuals of the VFM Y<sub>VFM</sub> component with respect to the Y<sub>VFM_Model</sub> value predicted by the initial VFM parent model in the VFM reference frame, and computing best linear fits of the same form as in Eq. (2):

$$ \Delta Y_{\text{VFM}} = \alpha_{\text{VFM}} Y_{\text{VFM\_Model}} + \beta_{\text{VFM}} \quad (3) $$

Figures S3 and S4 (analogous to Figures S1 and S2) in the Supporting Information provide plots of ΔY<sub>VFM</sub> as a function of Y<sub>VFM_Model</sub> for each of the LT and each of the Alpha and Bravo satellites. The recovered values of the α<sub>VFM</sub> slopes (and 2σ uncertainties) as a function of LT are summarized in Fig. 11b. Note that as the VFM dataset mirrors the ASM-V dataset, the data distribution as a
<figure><img src="image_14.png" /><figcaption>**Fig. 11** Values of the $\alpha_{ASM}$ (a) and $\alpha_{VFM}$ (b) slopes from, respectively, Eqs. (2) and (3), as a function of LT for the Alpha (red) and Bravo (blue) satellites, shown with their $2\sigma$ error bars ($\sigma$ being inferred from the least square best fit of Eqs. (2) and (3) to the data). Also shown as dotted lines (red for Alpha, blue for Bravo), the average $\bar{\alpha}_{ASM}$ and $\bar{\alpha}_{VFM}$ of these values (weighted by the amount of data available in each LT bin). Note that values for each LT T correspond to data selected between T and T + 1 h LT</figcaption></figure>

function of LT shown in Fig. 10 also applies to the VFM data used to produce Fig. 11b.

Both Fig. 11a and b reveals a clear dependence of the $\alpha_{ASM}$ and $\alpha_{VFM}$ slopes with the LT. This dependence is the same for Swarm Alpha and Bravo, showing that the ASM-V data on the one hand, and the VFM data on the other hand, behave in a similar way on both satellites. A common LT dependence is also found between the $\alpha_{ASM}$ and $\alpha_{VFM}$ slopes, with a clear maximum at 0 LT (deep in the night) and clear minima at 6 and 18 LT (dawn and dusk). In contrast, $\alpha_{ASM}$ and $\alpha_{VFM}$ values are clearly offset with respect to each other, with $\alpha_{ASM}$ displaying a positive average value $\bar{\alpha}_{ASM}$ (computed over all LT and weighted by the amount of data available in each LT bin) of order $1 \times 10^{-4}$ for both Alpha and Bravo, and $\alpha_{VFM}$ displaying a negative average value $\bar{\alpha}_{VFM}$ of order $-0.4 \times 10^{-4}$ for Alpha and $-0.3 \times 10^{-4}$ for Bravo.

## 6.2 Quantification of ASM and VFM instrument calibration errors using data residuals

To further interpret the above results, we first recall that when we directly compared ASM-V and VFM Y components in the instrument's reference frame, no systematic LT dependence of the type seen in the $\alpha_{ASM}$ and $\alpha_{VFM}$ slopes was found in the $\alpha$ slope between 17 and 7 LT (Fig. 6). For those LT, the $\alpha$ slope was consistently of order $\alpha = 2 \times 10^{-4}$. Since the vector components of the two ASM and VFM instruments are calibrated independently, this further suggests that, for at least those LT, the putative Y component calibration error on one or each of the two instruments is also independent of LT. These calibration errors can thus be expected to result in a constant contribution to the $\alpha_{ASM}$ and/or $\alpha_{VFM}$ slopes plotted in Fig. 11. This in turn would suggest that the LT dependence of these slopes is not related to a true calibration issue, but to some contribution of non-modeled

geophysical fields that remain in the $\Delta Y_{ASM}$ and $\Delta Y_{VFM}$ residuals. Such contribution can indeed be expected to be identical in both residuals, since the ASM and VFM should see the same non-modeled geophysical fields. This interpretation is strongly supported by the fact that both $\alpha_{ASM}$ and $\alpha_{VFM}$ slopes behave in the same way as a function of LT, and mainly differ by their average $\bar{\alpha}_{ASM}$ and $\bar{\alpha}_{VFM}$ values. Those $\bar{\alpha}_{ASM}$ and $\bar{\alpha}_{VFM}$ average value can thus be expected to be indicative of the true (LT independent) calibration errors affecting each of the ASM and VFM instruments. If that interpretation is correct, we should then expect the relative $\alpha$ calibration error, previously found when subtracting the ASM-V data from the VFM data in the instrument's reference frame, to be related to those $\bar{\alpha}_{ASM}$ and $\bar{\alpha}_{VFM}$ values through the simple relation $\alpha = \bar{\alpha}_{ASM} - \bar{\alpha}_{VFM}$. Indeed, the differences between $\bar{\alpha}_{ASM}$ and $\bar{\alpha}_{VFM}$ values, of order $1.4 \times 10^{-4}$ for Alpha and $1.3 \times 10^{-4}$ for Bravo are close to the $\alpha = 2 \times 10^{-4}$ order of magnitude found for both satellites in Sect. 5.

To consolidate this interpretation, we carry out a similar investigation on the X and Z components of both the ASM and VFM instruments, for which, based on our results in Sect. 5, no calibration issue is expected. Figure 12 shows the plots of the corresponding $\alpha_{ASM}$ and $\alpha_{VFM}$ slopes (and $2\sigma$ uncertainties) as a function of LT computed in the same way as in Fig. 11.

Recalling that $X_{ASM}$ is aligned with $Z_{VFM}$ while $Z_{ASM}$ is aligned with $X_{VFM}$ (Fig. 3), several important conclusions can be drawn from Fig. 12. We first again find no difference in behavior between the Alpha and Bravo satellites. We next find that the corresponding average values $\bar{\alpha}_{ASM}$ and $\bar{\alpha}_{VFM}$ are all within $\pm 5 \times 10^{-6}$, i.e., two orders of magnitude smaller than the average values $\bar{\alpha}_{ASM}$ and $\bar{\alpha}_{VFM}$ found for the $Y_{ASM}$ and $Y_{VFM}$ components. We finally find remarkably similar LT dependence of $\alpha_{ASM}$ for $X_{ASM}$ and $\alpha_{VFM}$ for $Z_{VFM}$ on the one hand (Fig. 12a and
<figure><img src="image_15.png" /><figcaption>Fig. 12 Same as Fig. 11, but for values of α<sub>ASM</sub> when considering the X<sub>ASM</sub> (a) and Z<sub>ASM</sub> (c) components and of α<sub>VFM</sub> when considering the Z<sub>VFM</sub> (b) and X<sub>VFM</sub> (d) components</figcaption></figure>

b), and of α<sub>ASM</sub> for Z<sub>ASM</sub> and α<sub>VFM</sub> for X<sub>VFM</sub> on the other hand (Fig. 12c and d)). These results support not only the fact that no calibration issue appears to affect the X and Z components of the ASM and VFM instruments, but also that all the LT dependency found in the α<sub>ASM</sub> and α<sub>VFM</sub> slopes for these components is likely the result of non-modeled geophysical fields. Finally, this also suggests that at least for those components, the contribution of non-modeled geophysical fields to ᾱ<sub>ASM</sub> and ᾱ<sub>VFM</sub> tend to nicely average out. Although this cannot be taken as a proof that the same applies to the ᾱ<sub>ASM</sub> and ᾱ<sub>VFM</sub> values recovered for the Y components of the ASM and VFM instruments, this further encourages us to use those average ᾱ<sub>ASM</sub> and ᾱ<sub>VFM</sub> values as first-order estimates of the impact of the miscalibration of those components.

Before proceeding, it is worth mentioning that we also checked the behavior of the β<sub>ASM</sub> and β<sub>VFM</sub> offsets jointly recovered with the various α<sub>ASM</sub> and α<sub>VFM</sub> discussed above. No systematic dependence with LT was found for these offsets, which always remain within 0.3 nT in absolute values, with an average value of less than 0.1 nT (often very less). Such offsets are significantly smaller than the β offsets we had found when directly comparing ASM-V and VFM data in the instrument's reference frame (Figs. 4 and 5), which we attributed to possible residual dBSun perturbations. We interpret this as a

consequence of the fact that in all the analysis carried out in the present section, no distinction was made between ascending and descending half-orbits, whereas, as previously noted, dBSun perturbations are sensitive not only to the LT, but also to the ascending/descending orientation of the spacecraft. This, we believe, likely smooths down contributions of any residual dBSun perturbations to the β<sub>ASM</sub> and β<sub>VFM</sub> offsets.

## 6.3 Iterative post-calibration protocol

If the ᾱ<sub>ASM</sub> and ᾱ<sub>VFM</sub> values recovered for the Y components are indeed indicative of the miscalibration of this component on each of the ASM and VFM instruments on Alpha and Bravo, a first-order strategy to mitigate this miscalibration is to use these values to simply rescale all Y components. However, rather than using the value ᾱ<sub>ASM</sub> and ᾱ<sub>VFM</sub> computed over all LT and weighted by the amount of data available in each LT bin, as we did so far, we finally compute a single final regression of the same form as Eqs. (2) and (3) using all residuals ΔY<sub>ASM</sub> and ΔY<sub>VFM</sub>, now making no LT distinction:

$$ \Delta Y_{\text{ASM}} = A_{\text{ASM}} Y_{\text{ASM\_Model}} + B_{\text{ASM}}, \quad (4) $$

$$ \Delta Y_{\text{VFM}} = A_{\text{VFM}} Y_{\text{VFM\_Model}} + B_{\text{VFM}} \quad (5) $$
<table><caption>**Table 2** Post-calibration parameters recovered from regression analysis of all residuals ΔY<sub>ASM</sub> and ΔY<sub>VFM</sub>, when no LT distinction is being made</caption><thead><tr><th></th><th>Alpha</th><th>Bravo</th></tr></thead><tbody><tr><td>A<sub>ASM</sub></td><td>+9.17350·10<sup>-5</sup> (0.08610·10<sup>-5</sup>)</td><td>+9.63282·10<sup>-5</sup> (0.08263·10<sup>-5</sup>)</td></tr><tr><td>A'<sub>ASM</sub></td><td>+3.76376·10<sup>-5</sup> (0.08578·10<sup>-5</sup>)</td><td>+3.89483·10<sup>-5</sup> (0.08232·10<sup>-5</sup>)</td></tr><tr><td>A<sub>VFM</sub></td><td>-4.28268·10<sup>-5</sup> (0.07019·10<sup>-5</sup>)</td><td>-3.32644·10<sup>-5</sup> (0.06967·10<sup>-5</sup>)</td></tr><tr><td>A'<sub>VFM</sub></td><td>-1.81616·10<sup>-5</sup> (0.07021·10<sup>-5</sup>)</td><td>-1.63305·10<sup>-5</sup> (0.06970·10<sup>-5</sup>)</td></tr></tbody></table>

A<sub>ASM</sub> and A<sub>VFM</sub> are the parameters used for the first post-calibration of, respectively, ASM-V and VFM data; A'<sub>ASM</sub> and A'<sub>VFM</sub> are the analogous parameters used for the second post-calibration. Each value is independently provided for each Alpha and Bravo satellites. All parameters are provided with their 1 σ uncertainty in brackets

This is done independently on Alpha and Bravo. The corresponding four regressions are illustrated in Figures S5. These regressions lead to the A<sub>ASM</sub> and A<sub>VFM</sub> values (and 1 σ uncertainties) shown in Table 2. Using these values, we next rescale all Y<sub>ASM</sub> and Y<sub>VFM</sub> data with the appropriate factor for each instrument and satellite, leading to the intermediate values (1 − A<sub>ASM</sub>)Y<sub>ASM</sub> and (1 − A<sub>VFM</sub>)Y<sub>VFM</sub>. As such a rescaling would no longer make the modulus of the vector data consistent with the matching absolute scalar value F originally used for calibrating the data, we next also renormalize each vector data by the matching scalar data. This process leads to what we define as post-calibrated X', Y', Z' data of the form:

$$
\begin{align}
X' &= (X^2 + (1-A)^2 Y^2 + Z^2)^{-\frac{1}{2}} F \times X \nonumber \\
Y' &= (1-A)(X^2 + (1-A)^2 Y^2 + Z^2)^{-\frac{1}{2}} F \times Y, \tag{6} \\
Z' &= (X^2 + (1-A)^2 Y^2 + Z^2)^{-\frac{1}{2}} F \times Z, \nonumber
\end{align}
$$

where X, Y and Z are the original vector components, F the matching scalar data and A the appropriate A<sub>ASM</sub> or A<sub>VFM</sub> value for each satellite.

To test the improvement brought by this post-calibration strategy, we recompute a new set of twin parent models using exactly the same procedure and parameters as the initial twin parent models. We also use exactly the same selection of data as previously, and simply use the post-calibrated data in place of the original data. We next repeat the residual analysis that led to Fig. 11. The corresponding results are shown in Fig. 13a and b. Comparing these to Fig. 11a and b shows that the post-calibration data led to residuals displaying very similar dependence to LT but with reduced α<sub>ASM</sub> and α<sub>VFM</sub> values, with α<sub>ASM</sub> now of order 0.4 × 10<sup>-4</sup> for both Alpha and Bravo, and α<sub>VFM</sub> of order −0.16 × 10<sup>-4</sup> for Alpha and −0.14 × 10<sup>-4</sup> for Bravo. This is exactly what one would have expected if the post-calibration goes some way at correcting the

<figure><img src="image_16.png" /><figcaption>**Fig. 13** Same as Fig. 11, but for values of α<sub>ASM</sub> and α<sub>VFM</sub> after first post-calibration (a, b) and after second post-calibration (c, d)</figcaption></figure>
original Y calibration issue. These results, however, show that this first attempt of post-calibration did not fully remove the original Y calibration issue. The reason for this is that in our approach to compute the A<sub>ASM</sub> or A<sub>VFM</sub> values, we assumed the initial parent field models to faithfully predict the expected Y field component. This ignored the fact that even though the models tended to reject the excess (or default) Y component value to match the constraints correctly provided by the well-calibrated X and Z components, the optimization problem can take advantage of some leeway to accommodate some of the excess (or default) on the Y component. This can then be expected to lead to ΔY<sub>ASM</sub> and ΔY<sub>VFM</sub> residuals with underestimated miscalibration contributions.

To improve the post-calibration, we take advantage of the possibility of iterating the above procedure one more step. New A<sub>ASM</sub>' and A<sub>VFM</sub>' values can be computed in exactly the same way, now using the post-calibrated X', Y', Z' data and corresponding new parent models in place of the initial data and parent models in Eqs. (4) and (5). The corresponding four regressions are illustrated in Figures S6. These regressions lead to the A<sub>ASM</sub>' and A<sub>VFM</sub>' values (and 1 σ uncertainties) also provided in Table 2. They can then be used to better rescale the initial Y components through (1 − A'<sub>ASM</sub>)(1 − A<sub>ASM</sub>) Y<sub>ASM</sub> and (1 − A'<sub>VFM</sub>)(1 − A<sub>VFM</sub>) Y<sub>VFM</sub> and next produce a new (and final) set of post-calibrated X'', Y'', Z'' data using Eq. (4), where we now replace all (1 − A) factors by (1 − A')(1 − A). These data are finally used to compute a new (and final) set of twin parent models. Figure 13c and d indeed shows that this leads to the expected additional improvement. The ᾱ<sub>ASM</sub> and ᾱ<sub>VFM</sub> values are now very close to zero, with ᾱ<sub>ASM</sub> down to 0.2 × 10<sup>−4</sup> for both Alpha and Bravo, and ᾱ<sub>VFM</sub> down to −0.06 × 10<sup>−4</sup> for Alpha and −0.05 × 10<sup>−4</sup> for Bravo. This was considered as successful enough and we did not attempt to iterate the process once more, given the crude nature of this post-processing protocol. Important to further mention is that the renormalization of the X and Z components involved in this procedure (through Eq. 6) is extremely minor and that reproducing the analysis that led to Fig. 12 with the renormalized X'' and Z'' components provides virtually indistinguishable results. It does not alter our previous conclusion that no calibration issue appears to affect the X and Z components of the ASM and VFM instruments.

## 7 Final twin parent models

Following the analysis carried out in the previous section, the final twin parent models we produced are those computed using the final post-calibrated X'', Y'', Z'' data. No new data selection was introduced. We relied on exactly the same data distribution, just replacing the original

<table><caption>**Table 3** Residual statistics for all data used to produce the final ASM-V and VFM twin models. Same conventions as in Table 1</caption><thead><tr><th></th><th>N</th><th colspan="2">ASM-V (ASM-V data)</th><th colspan="2">VFM twin (VFM data)</th></tr><tr><th></th><th></th><th>Mean (nT)</th><th>RMS (nT)</th><th>Mean (nT)</th><th>RMS (nT)</th></tr></thead><tbody><tr><td colspan="6"><em>Alpha + Bravo</em></td></tr><tr><td>F</td><td>612360</td><td>-0.06</td><td>3.81</td><td>-0.04</td><td>3.81</td></tr><tr><td>F and B<sub>B</sub></td><td>3019173</td><td>0.01</td><td>2.50</td><td>0.02</td><td>2.48</td></tr><tr><td>B<sub>B</sub></td><td>2406813</td><td>0.03</td><td>2.10</td><td>0.03</td><td>2.07</td></tr><tr><td>B<sub>r</sub></td><td>2406813</td><td>-0.02</td><td>2.18</td><td>0.00</td><td>1.54</td></tr><tr><td>B<sub>θ</sub></td><td>2406813</td><td>0.02</td><td>3.24</td><td>0.03</td><td>2.77</td></tr><tr><td>B<sub>φ</sub></td><td>2406813</td><td>0.00</td><td>2.83</td><td>0.00</td><td>2.46</td></tr><tr><td colspan="6"><em>Alpha</em></td></tr><tr><td>F</td><td>302426</td><td>-0.14</td><td>3.95</td><td>-0.12</td><td>3.95</td></tr><tr><td>F and B<sub>B</sub></td><td>1475747</td><td>0.00</td><td>2.56</td><td>-0.01</td><td>2.54</td></tr><tr><td>B<sub>B</sub></td><td>1173321</td><td>0.03</td><td>2.13</td><td>0.02</td><td>2.10</td></tr><tr><td>B<sub>r</sub></td><td>1173321</td><td>0.01</td><td>2.27</td><td>0.01</td><td>1.56</td></tr><tr><td>B<sub>θ</sub></td><td>1173321</td><td>0.00</td><td>3.28</td><td>0.05</td><td>2.78</td></tr><tr><td>B<sub>φ</sub></td><td>1173321</td><td>0.00</td><td>2.87</td><td>0.00</td><td>2.46</td></tr><tr><td colspan="6"><em>Bravo</em></td></tr><tr><td>F</td><td>309934</td><td>0.02</td><td>3.68</td><td>0.04</td><td>3.68</td></tr><tr><td>F and B<sub>B</sub></td><td>1543426</td><td>0.03</td><td>2.44</td><td>0.04</td><td>2.42</td></tr><tr><td>B<sub>B</sub></td><td>1233492</td><td>0.03</td><td>2.06</td><td>0.04</td><td>2.04</td></tr><tr><td>B<sub>r</sub></td><td>1233492</td><td>-0.04</td><td>2.10</td><td>-0.02</td><td>1.51</td></tr><tr><td>B<sub>θ</sub></td><td>1233492</td><td>0.05</td><td>3.20</td><td>0.01</td><td>2.76</td></tr><tr><td>B<sub>φ</sub></td><td>1233492</td><td>-0.01</td><td>2.79</td><td>0.00</td><td>2.45</td></tr></tbody></table>

ASM-V or VFM vector data by the matching final post-processed vector data. The parameterization and damping parameter choices were also unchanged. These final parent models thus only differ from the initial parent models because we now use the final post-calibrated data. Table 3 provides the corresponding residual statistics in the same format as in Table 1. Comparing Tables 3 to 1 shows that these statistics are very similar to those of the initial twin models. Worth noting, however, is a quasi-systematic (very) slight improvement in the parameters of the statistics of the final ASM-V parent model compared to that of the initial ASM-V parent model. Some improvement can also be seen in many parameters of the statistics of the final VFM twin model compared to that of the initial VFM twin model. This is a faint but positive sign that the post-calibrated data did not degrade the modeling, and may have improved it.

Best evidence of the success of our post-calibration strategy is provided in the maps of Fig. 14. Figure 14c displays the differences between the predictions of the final ASM-V and VFM twin parent models for the B<sub>φ</sub> component. It shows that much of the differences previously seen in the analogous map of Fig. 2c for the predictions of the initial ASM-V and VFM twin parent are reduced.
<figure><img src="image_17.png" /><figcaption>**Fig. 14** Differences between the predictions of the final ASM-V and VFM twin parent models for the B<sub>r</sub> (a), B<sub>θ</sub> (b) and B<sub>ψ</sub> (c) components. Also shown in d, the difference between the predictions of the final ASM-V and the official DGRF2020 for the B<sub>r</sub> component (note the factor two in the color scale compared to the other maps). All models are computed up to SH degree and order 13, and all maps are plotted at Earth's surface (6371.2 km) for epoch 2020.0</figcaption></figure>

The minimum and maximum values (-0.432 nT and 0.577 nT) are down by a factor two (previously -0.992 nT and 1.116 nT). The same holds for the standard deviations computed over the two maps (previously 0.383 nT, now 0.178 nT). Hardly any of the common large-scale structure seen in Fig. 2c and d that prompted our post-calibration approach is left.

Figure 14a and b displays maps of the differences between the predictions of the final ASM-V and VFM twin parent models for the B<sub>r</sub> and B<sub>θ</sub> components. Comparing these maps to the analogous maps of Fig. 2a and b further shows that the previously observed improvement in the B<sub>ψ</sub> component did not come at the expense of some increased disagreement between the B<sub>r</sub> and B<sub>θ</sub> predictions of the two models. The minimum and maximum values in the B<sub>r</sub> maps (-1.165 nT and 1.341 nT) are essentially unchanged (previously -1.424 nT and 1.140 nT), but the standard deviation is down from 0.606 nT to 0.497 nT. Minimum, maximum and standard deviation values in the B<sub>θ</sub> maps (-0.865 nT, 0.909 nT and 0.317 nT) are all down (previously -1.177 nT, 0.988 nT and 0.380 nT). This shows that correcting for the calibration issues in the Y components of the ASM-V and VFM instruments succeeded at improving agreement between predictions of

the models not only for the B<sub>ψ</sub> component as one would have expected, but also more generally on all components. This improvement is found throughout the entire period covered by the two final parent models (April 2014–July 2024).

Figure 15 provides a useful final complementary spectral illustration of this improvement. We use the classical Lowes–Mauersberger SH spectrum (Mauersberger 1956; Lowes 1974) and plot such spectra for the differences between the initial ASM-V and VFM parent models (blue dotted line) and for the final ASM-V and VFM parent models (red dotted line). We focus here on epoch 2020.0 as this is the most relevant epoch given the ultimate purpose of this study to produce a DGRF2020 candidate model. As can be seen, the difference between the final ASM-V and VFM parent models always leads to spectral values below those of the initial ASM-V and VFM parent models, except marginally so for SH degrees 10 and 11. Most significant improvements are seen in the SH degrees 1, 6, and 8, commensurate with the improvements seen in the maps of Fig. 14. The largest disagreement between the final ASM-V and VFM parent models is found for SH degree 3, and is less than 0.2 nT<sup>2</sup>, with all other SH degrees well below 0.1 nT<sup>2</sup>.
<figure><img src="image_18.png" /><figcaption>**Fig. 15** Lowes-Mauersberger spherical harmonic spectra (Mauersberger 1956; Lowes 1974) of the final ASM-V parent model (black solid line), of the difference between the final ASM-V and VFM twin parent models (red dotted line), of the difference between the initial ASM-V and VFM twin parent models (blue dotted line), of the difference between the final ASM-V parent and official DGRF2020 models (black dotted line), and of the difference between the initial ASM-V parent and official DGRF2020 models (green dotted line). All spectra are plotted at Earth's surface (6371.2 km) for epoch 2020.0</figcaption></figure>

## 8 DGRF2020 candidate model and associated error estimates

Given the success of our post-calibration strategy, the DGRF2020 candidate model we sent in response to the IGRF-14 call of the Working Group V-MOD of IAGA is simply extracted from our final ASM-V parent model for epoch 2020.0, up to SH degree and order 13. This model is entirely and only based on ASM data. The post-calibration we used to mitigate the Y component calibration issue we identified is also entirely and only based on ASM data. Indeed, although our investigation was greatly helped by the comparison of ASM-V and VFM twin parent models and data, the post-calibration approach we described in Sect. 6 is implemented completely independently on the ASM-V and VFM data.

To derive error estimates associated with our DGRF2020, as requested by the IGRF-14 call, we nevertheless took advantage of the availability of the two post-processed ASM-V and VFM datasets. We just followed the approach of Vigneron et al. (2021).

We first split the post-calibrated ASM-V data set in two. This is done by ordering and numbering data chronologically, and placing even-numbered data in a first sub-dataset ASM-V-1, and every odd-numbered data in the second sub-dataset ASM-V-2. Likewise, the twin VFM data set (post-calibrated Swarm L1b nominal data, version 0602/0606) is split in two VFM-1 and VFM-2 datasets (the time distribution of VFM-1/2 data matching

that of ASM-V-1/2). A model is next computed from each sub-dataset, using the same parameters as the final parent and twin models, leading to four models: ASMV-1 and VFM-1 (sharing the same even-numbered data distribution) and ASMV_2 and VFM_2 (sharing the same odd-numbered data distribution). The values A1 (from ASMV_1), A2 (from ASMV-2), V1 (from VFM1) and V2 (from VFM2) of each Gauss coefficient are next used to compute the quantity:

$$ \sigma = \sqrt{\frac{(A_1 - V_2)^2 + (A_2 - V_1)^2}{2}}, \qquad (5) $$

which we assign as a 1 σ type of error to the corresponding Gauss coefficient. These values were provided with 2 significant digits.

Note that this approach only provides estimates of standard deviations and ignores possible cross-correlations between Gauss coefficients. It also only assesses uncertainties produced by the quality of the data (and not those related to the choice of the modeling strategy, the choice of data selection criteria, or the choice of parameters for the inversion). These errors should thus be viewed as a lower bound indication of the real errors of the overall errors affecting the model. Nevertheless, they are consistent with the observed spectral difference between the final twin parent models as plotted in Fig. 15.

## 9 DGRF2020 candidate model a posteriori performance

A simple way of a posteriori assessing the performance of our DGRF2020 candidate model is to compare it to the official DGRF2020, which was later produced and finally adopted by the IAGA Working Group V-MOD. By construction, this model was not available at the time we produced our candidate model. This assessment is done here in two ways.

We first show in Fig. 14d a map of the difference between the predictions of the final ASM-V parent model for epoch 2020.0 (hence of our DGRF2020 candidate model) and of the official DGRF2020 for the B$_r$ component. This map shows that both models agree slightly less than the two final ASM-V and VFM parent models (Fig. 14a), but now mainly at small scales and with maximum disagreements at high (mainly South) latitudes. The minimum and maximum values are, respectively, of -2.606 nT and 3.632 nT (compared to -1.165 nT and 1.341 nT in Fig. 14a), and the standard deviation computed over the map is of now 0.752 nT (compared to 0.497 nT in Fig. 14a). Similar maps computed by Beggan et al. (2025b) for all 14 DGRF2020 candidate models (their Fig. 3) show that our DGRF candidate model
is among the very few models closest to the official DGRF2020 model.

We next show in Fig. 15 (black dotted line), the SH spectral comparison of our DGRF2020 candidate model (final ASM-V parent model for epoch 2020.0) with the official DGRF2020 model. The agreement is again excellent, with a maximum disagreement of less than 0.2 nT² for SH degree 3 and disagreements for all other SH degrees well below 0.1 nT². Similar and more detailed spectral comparisons carried out by Beggan et al. (2025b) for all 14 DGRF2020 candidate models (their Figs. 1 and 2) confirm that our DGRF candidate model is among the very few models closest to the official DGRF2020 model.

To further assess the benefit of the post-calibration, we introduced for the production of our DGRF2020 candidate model, we plot the SH comparison between the initial ASM-V parent model for epoch 2020 with the official DGRF2020 model in Fig. 15 (green dotted line). This initial ASM-V parent model is not corrected for the Y component calibration issue on the ASM instrument. This model always performs worse than our final DGRF2020 model (green dotted line always above black dotted lines) for all SH degrees between 1 and 6, and equally well for SH degrees between 7 and 13. This is extremely encouraging and confirms that our post-calibration of the ASM-V data was beneficial. Important to recall, indeed, is that none of the other DGRF2020 candidate models relied on any ASM-V data, and that the official DGRF2020 model constructed by using all 14 DGRF2020 candidates is itself only very mildly influenced by the ASM-V data through our own DRGF2020 candidate model. Quite remarkable in this respect is the fact that our post-calibration strategy led to even better agreement at large scales between our DGRF2020 candidate model and the official DGRF2020 model than between our final ASM-V and VFM twin models. This can already be seen in Fig. 14, but is best seen in the spectra of Fig. 15 (black dotted curve matching or clearly below the red dotted curve up to SH degree 5). This was achieved even though no other candidate model implemented any post-calibration on the VFM data they used (for those who used Swarm data). This final result can possibly be interpreted in terms of some remaining perturbations still affecting both the ASM-V or the VFM data sets, but in different ways, which the dBSun corrections and our post-calibration strategy could not remove from the ASM-V and VFM data, thus affecting independently the ASM-V and VFM twin models. In contrast, as the official DGRF model is being produced from DGRF candidate models that did not rely on the same modeling, data sets, and data selection scheme, it may be that the merging procedures used (by taking the median of the Gauss coefficients of all the candidate models, see Beggan et al. (2025a)) did succeed

at reducing the impact of these remaining ASM-V and VFM data perturbations on the official DGRF model. If that is the case, this could be the reason why our final ASM-V model (still affected by some remaining ASM perturbations) is in better agreement at large scales with the official DGRF model (with less impact of remaining ASM and VFM perturbations) than with the VFM twin model (still affected by remaining VFM perturbations).

## 10 Concluding comments

Beyond demonstrating that yet another excellent, and improved, DGRF candidate model could be built using Swarm ASM-V vector data, the present study again highlights the benefit of having two magnetometers capable of simultaneously providing independent vector data on board the Swarm satellites to identify and mitigate issues affecting the data. This allowed us to uncover the latest issue identified so far, the occurrence of a small, but significant miscalibration in the Y components of both the ASM-V and VFM vector data. We implemented a method to independently quantify this miscalibration on both instruments and designed a post-calibration strategy which we successfully implemented to produce further improved ASM-V and VFM vector data. This method was implemented independently on each of the ASM-V and VFM data sets. It could also be used to carry on similar analysis and corrections on any other satellite mission with absolute magnetometry, even if such missions only carry the standard combination of just one vector magnetometer combined with one absolute scalar magnetometer.

An important question that our study raises, however, is the reason why the in-flight procedures nominally used for calibrating the ASM-V and VFM vector data produce such miscalibration in the Y components, and not in the other X and Z components. Since the Swarm satellites are polar orbiting, it is tempting to speculate that the underlying cause is that the Y components (roughly aligned with the $B_\psi$ main field component) is the one experiencing lesser values (compared to the X and Z components). Indeed, only in-flight data are used to calibrate the instruments and both calibration optimization procedures used for the VFM (Tøffner-Clausen et al. 2016) and the ASM-V data (Gravrand et al. 2001; Vigneron et al. 2021) involve minimizing the residual between the modulus of the vector measurements and the reference absolute scalar field measurement. It may thus well be that the recovered calibration parameters are biased by the biased in-flight calibration data set.

Tests on the ASM instrument could be carried out on ground by using a spare ASM instrument and reproducing the calibration procedure in a controlled set up, exposing the instrument to ground simulated orbital
data and comparing the recovered calibration parameters to the parameters recovered when exposing the instrument to an ideal isotropic (equivalent for all components) dataset. Demonstrating this effect on the VFM instrument would probably be more difficult, especially that the in-flight calibration procedure used for that instrument is more elaborate and further involves co-estimating the dBSun perturbation. This, we further note, may well be the reason why the (smaller) miscalibration found in the Y component of the VFM data is in the opposite direction (producing smaller estimates of Y than expected) compared to that found in the Y component of the ASM data (producing larger estimates than expected).

Such tests could also help improve in-flight calibration procedures to avoid having to rely on the useful, but rough, empirical post-calibration procedure we introduced. This would be particularly useful for forthcoming missions, such as the NanoMagSat mission to be launched end of 2027, which will rely on a minaturized version of the ASM to produce absolute vector data (Hulot et al. 2018; Deconinck et al. 2025; Marchese et al. 2025).

### Abbreviations

ASM Absolute scalar magnetometer

ASM-V ASM vector mode data

CCDB Characterization and calibration data base

CEA-Leti Commissariat à l'Energie Atomique et aux énergies alternatives, Laboratoire d'Electronique et de Technologie de l'Information

CFI Customer furnished instrument

CNES Centre National d'Etudes Spatiales

CRF Central reference frame

CSES China seismo-electromagnetic satellite

DGRF Definitive geomagnetic reference field

DTU Technical University of Denmark

ELF Extremely low frequencies

ESA European Space Agency

GSM Geocentric solar magnetospheric (coordinate system)

IAGA International association of geomagnetism and aeronomy

IGRF International geomagnetic reference field

LT Local time

MLI Multi-layer insulation

MSS-1 Macau scientific satellite 1

QD Quasi-dipole

SH Spherical harmonic

SM Solar magnetic (coordinate system)

STR Star trackers

VFM Vector flux magnetometer

### Supplementary Information

The online version contains supplementary material available at https://doi.org/10.1186/s40623-025-02343-1.

Additional file1 (DOCX 24 kb)

### Acknowledgements

The authors wish to thank O. Bonnot (Geomagnetism team of IPGP) for his technical support in producing ASM-V data and making them publicly available. The authors also wish to thank the two anonymous reviewers for their very helpful comments on the original draft of this paper.

### Author contributions

GH, PI of the ASM instruments, designed the study, carried out the analysis, conceived the calibration investigation and post-calibration strategy, and wrote the manuscript; LC carried out data selection, led model computations, produced all figures and tables and participated in all steps of the study; RD processed all ASM-V data used in this study and contributed to model computations; JML led the design of the ASM instruments and its processing chain up to L1a data together with TJ, who also computed and provided the CCDB files; MF contributed to the calibration investigation and, together with PC, to the DGRF2020 candidate model assessment. All authors read and approved the final manuscript.

### Funding

This study was supported by the Centre National d'Etudes Spatiales (CNES) through the project "Suivi et exploitation de la mission Swarm", which also funds the position of RD and the production of ASM-V data.

### Availability of data and materials

The experimental ASM-V data used in this study are ASM-V data version 060X (0602, 0605, or 0606) produced by IPGP/CEA-Léti using the ASM CEA-Léti designed/ CNES Customer Furnished Instrument. These ASM-V data are accessible at https://doi.org/10.18715/IPGP.2025.m7oic39m for Alpha and https://doi.org/10.18715/IPGP.2025.m7ok4rm for Bravo. Note that the format of these ASM-V data not only provides the ASM-V vector data (in the ASM reference frame), but also directly the ASM scalar data, the VFM vector data (in the VFM reference frame), and the q NEC_CRF quaternions used in this study. Additional ASMxAUX data version 060X (0602, 0605, or 0606) also used in this study are available at https://earth.esa.int/eogateway/missions/swarm/data. The final DGRF 2020 candidate model with associated error estimates (together with all IGRF-14 candidate models) is accessible at the IAGA-VMOD GitHub repository https://github.com/IAGA-VMOD/IGRF14eval/tree/main/docs. The final ASM-V parent model is accessible at https://doi.org/10.18715/IPGP.2025.mhoxy84p.

### Declarations

#### Ethics approval and consent to participate

Not applicable.

#### Consent for publication

Not applicable.

#### Competing interests

The authors declare that they have no competing interest.

#### Author details

¹Institut de Physique du Globe de Paris, CNRS, Université Paris Cité, 75005 Paris, France. ²Département Système, CEA-Leti, Université Grenoble Alpes, MINATEC 17, rue des Martyrs, 38054 Grenoble Cedex 9, France.

Received: 28 August 2025 Accepted: 2 December 2025

Published online: 14 January 2026

### References

Alken P, Thébault E, Beggan C, Amit H, Aubert J, Baerenzung J, Bondar TN, Brown W, Cali S, Chambodut A, Chulliat A, Cox G, Finlay CC, Fournier A, Gillet N, Grayver A, Hammer M, Holschneider M, Huder L, Hulot G, Jager T, Kloss C, Korte M, Kuang W, Kuvshinov A, Langlais B, Léger J-M, Lesur V, Livermore PW, Lowes FJ, Macmillan S, Magnes W, Mandea M, Marsal S, Matzka J, Metman MC, Minami T, Morschhauser A, Mound JE, Nair M, Nakano S, Olsen N, Pavon-Carrasco FJ, Petrov VG, Ropp G, Rother M, Sabaka TJ, Sanchez S, Saturnino D, Schnepf N, Shen X, Stolle C, Tangborn A, Töffner-Clausen L, Toh H, Torta JM, Varner J, Vigneron P, Vervelidou F, Wardinski I, Wicht J, Woods A, Yang Y, Zeren Z, Zhou B (2021a) International geomagnetic reference field: the thirteenth generation. Earth Planets Space 73:49. https://doi.org/10.1186/s40623-020-01288-x

Alken P, Thébault E, Beggan C, Aubert J, Baerenzung J, Brown W, Califf S, Chulliat A, Cox G, Finlay CC, Fournier A, Gillet N, Hammer MD, Holschneider M,
Hulot G, Korte M, Lesur V, Livermore P, Lowes F, Macmillan S, Nair M, Olsen N, Ropp G, Rother M, Schnepf NR, Stolle C, Toh H, Vervelidou F, Vigneron P, Wardinski I (2021b) Evaluation of candidate models for the 13th international geomagnetic reference field. Earth Planets Space 73:48. https://doi.org/10.1186/s40623-020-01281-4

Beggan CD et al (2025a) International geomagnetic reference field: the fourteenth generation. Earth Planets Space (in press)

Beggan CD et al (2025b) Evaluation of candidate models for the 13th generation international geomagnetic reference field. Earth Planets and Space (submitted)

Brauer P, Töffner-Clausen L, Hulot G (2024) Review of the magnetic perturbation model of the Swarm Satellites, Swarm 10 year Anniversary & Science Conference, 08–12 April 2024, Copenhagen, Denmark, https://airdrive.eventsair.com/eventsairwesteuprod/production-nikal-public/4213d129d2874d048b74bd42d52c6885

Coïsson P, Chauvet L, Jenner P, Deborde R, Hulot G, Madelon R (2025) Global occurrences of whistlers detected in the Extremely Low Frequencies during Absolute Scalar Magnetometer burst mode acquisition campaigns of the Swarm mission. J Space Weather Space Clim https://doi.org/10.1051/swsc/2025048

Deconinck F, Hulot G, Léger J-M, Clausen LBN, Pastena M, Lejauffet J-P, Wilkinson J, Mataix Caballero D, Vallmitjana Tostado M, Coisson P, Jager T, Nieto P, Jørgensen JL (2025) NanoMagSat Status: a 3X16U Low-Earth Orbiting constellation to monitor the Earth magnetic field and the ionospheric environment. In: Proceedings volume of small satellites systems and services symposium (4S 2024), 135462K. https://doi.org/10.1117/12.3062471

Emsley AJ, Beggan CD, Callow A, Whaler K.A, Coisson P, Chauvet L, Hulot G, Jenner M (2025) Swarm absolute scalar magnetometer burst mode: observed elf signals and their origins. J Space Weather Space Clim https://doi.org/10.1051/swsc/2025048

Fillion M, Hulot G, Alken P, Chulliat A (2023) Modelling the climatology of low- and mid-latitude F-region ionospheric currents using the Swarm constellation. J Geophys Res Space Phys 128:e2023JA031344. https://doi.org/10.1029/2023JA031344

Finlay CC, Kloss C, Olsen N, Hammer MD, Töffner-Clausen L, Grayver A, Kuvshinov A (2020) The CHAOS-7 geomagnetic field model and observed changes in the South Atlantic Anomaly. Earth Planets Space 72:156. https://doi.org/10.1186/s40623-020-01252-9

Fournier A, Aubert J, Lesur V, Ropp G (2021) A secular variation candidate model for IGRF-13 based on Swarm data and ensemble inverse geodynamo modelling. Earth Planets Space 73:43. https://doi.org/10.1186/s40623-020-01309-9

Fournier A, Aubert J, Lesur V (2025) A secular variation candidate model for IGRF-14 combining observatory and satellite data with ensemble inverse geodynamo modelling. Technical note, https://github.com/IAGA-VMOD/IGRF14eval/blob/main/docs/IGRF14_IPGP_SV_doc.pdf

Fratter I, Léger J-M, Bertrand F, Jager T, Hulot G, Brocco L, Vigneron P (2016) Swarm absolute scalar magnetometers first in-orbit results. Acta Astronaut 121:76–87. https://doi.org/10.1016/j.actaastro.2015.12.025

Friis-Christensen E, Lühr H, Hulot G (2006) Swarm: a constellation to study the Earth's magnetic field. Earth Planets Space 58:351–358. https://doi.org/10.1186/BF03351933

Gravrand O, Khokhlov A, Le Mouël J-L, Léger J-M (2001) On the calibration of a vectorial 4He pumped magnetometer. Earth Planets Space 53:949–958. https://doi.org/10.1186/BF03351692

Holme R, Bloxham J (1996) The treatment of attitude errors in satellite geomagnetic data. Phys Earth Planet Inter 98:221–233. https://doi.org/10.1016/S0031-9201(96)03189-5

Hulot G, Vigneron P, Léger J-M, Fratter I, Olsen N, Jager T, Bertrand F, Brocco L, Sirol O, Lalanne X, Boness A, Cattin V (2015) Swarm's absolute magnetometer experimental vector mode, an innovative capability for space magnetometry. Geophys Res Lett. https://doi.org/10.1002/2014GL062700

Hulot G, Léger J-M, Vigneron P, Jager T, Bertrand F, Coisson P, Deram P, Boness A, Tomasini L, Faure B (2018) Nanosatellite high-precision magnetic mission enabled by advances in a stand-alone scalar/vector absolute magnetometer. In: IGARSS 2018—2018 IEEE international geoscience and remote sensing symposium, 978-1-5386-7150-4/18/$31.00, pp 6324–6327. https://doi.org/10.1109/IGARSS.2018.8517754

Jenner M, Coisson P, Hulot G, Buresova D, Truhlik V, Chauvet L (2024) Total root electron content: a new metric for the ionosphere below low earth

orbiting satellites. Geophys Res Lett 51:e2024GL110559. https://doi.org/10.1029/2024GL110559

Léger J-M, Bertrand F, Jager T, Prado ML, Fratter I, Lalaurie JC (2009) Swarm absolute scalar and vector magnetometer based on helium 4 optical pumping. Proc Chem 1:634–637. https://doi.org/10.1016/j.proche.2009.07.158

Léger J-M, Jager T, Bertrand F, Hulot G, Brocco L, Vigneron P, Lalanne X, Chulliat A, Fratter I (2015) In-flight performance of the Absolute Scalar Magnetometer vector mode on board the Swarm satellites. Earth Planets Space 67:57. https://doi.org/10.1186/s40623-015-0231-1

Lesur V, Shi Y (2024) IPGP main field candidate for IGRF 2020. Technical note. https://github.com/IAGA-VMOD/IGRF14eval/blob/main/docs/IPGP_IGRF2025_doc.pdf

Lowes FJ (1974) Spatial power spectrum of the main geomagnetic field, and extrapolation to the core. Geophys J Int 36(3):717–730. https://doi.org/10.1111/j.1365-246X.1974.tb00622.x

Macmillan S, Finlay C (2011) The international geomagnetic reference field. In: Mandea M, Korte M (eds) Geomagnetic observations and models, vol 5. IAGA Special Sopron Book Series. Springer, Heidelberg, pp 265–276

Marchese V, Lecuyot A, Rodrigues G, Arranz CJ, Deconinck F, Vallmitjana Tostado M, Wilkinson J, Hulot G, Léger J-M, Jager T, Nieto P, Fayos J, Algarra P, Pipo A, Zubeldia X (2025) Risk retirement activity for the NanoMagSat mission concept. In: Proceedings volume of small satellites systems and services symposium (4S 2024), 135460P. https://doi.org/10.1117/12.3061022

Mauersberger P (1956) Das Mittel der Energiedichte des geomagnetischen Hauptfeldes an der Erdoberfläche und seine säkulare Änderung. Gerlands Beitr Geophys 65:207–215

Merayo JMG, Jørgensen JL, Friis-Christensen E, Brauer P, Primdahl F, Jørgensen PS, Allin TH, Denver T (2008) The Swarm magnetometry package. In: Sandau R, Röser HP, Valenzuela A (eds) Small satellites for earth observation. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-6943-7_13

Olsen N, Friis-Christensen E, Floberghagen R, Alken P, Beggan CD, Chulliat A, Doornbos E, da Teixeira Encarnação J, Hamilton B, Hulot G, van den Ijsdel J, Kuvshinov A, Lesur V, Lühr H, Macmillan S, Maus S, Noja M, Olsen PEH, Park J, Plank G, Püthe C, Rauberg J, Ritter P, Rother M, Sabaka TJ, Schachtschneider R, Sirol O, Stolle C, Thébault E, Thomson AWP, Töffner-Clausen L, Velimsky J, Vigneron P, Visser PN (2013) The swarm satellite constellation application and research facility (SCARF) and swarm data products. Earth Planets Space 65(11):1189–1200. https://doi.org/10.5047/eps.2013.07.001

Olsen N, Lühr H, Finlay CC, Sabaka TJ, Michaelis I, Rauberg J, Töffner-Clausen L (2014) The CHAOS-4 geomagnetic field model. Geophys J Int 197:815–827. https://doi.org/10.1093/gji/ggu033

Ropp G, Lesur V (2023) Mid-latitude and equatorial core surface flow variations derived from observatory and satellite magnetic data. Geophys J Int 234:1191–1204. https://doi.org/10.1093/gji/ggad113

Ropp G, Lesur V, Baerenzung J, Holschneider M (2020) Sequential modelling of the Earth's core magnetic field. Earth Planets Space 72:153. https://doi.org/10.1186/s40623-020-01230-1

Thébault E, Finlay CC, Beggan C, Alken P, Aubert J, Barrois O, Bertrand F, Bondar T, Boness A, Brocco L, Canet E, Chambodut A, Chulliat A, Coisson P, Civet F, Du A, Fournier A, Fratter I, Gillet N, Hamilton B, Hamoudi M, Hulot G, Jager T, Korte M, Kuang W, Lalanne X, Langlais B, Léger JM, Lesur V, Lowes FJ, Macmillan S, Mandea M, Manoj C, Maus S, Olsen N, Petrov V, Rother M, Sabaka TJ, Saturnino D, Schachtschneider R, Visser O, Tangborn A, Taylor V, Thomson A, Töffner-Clausen L, Vigneron P, Wardinski I, Zvereva T (2015a) International geomagnetic reference field: the twelfth generation. Earth Planets Space 67:79. https://doi.org/10.1186/s40623-015-0228-9

Thébault E, Finlay CC, Alken P, Beggan CD, Canet E, Chulliat A, Langlais B, Lesur V, Lowes FJ, Manoj C, Rother M, Schachtschneider R (2015b) Evaluation of candidate geomagnetic field models for IGRF-12. Earth Planets Space 67:112. https://doi.org/10.1186/s40623-015-0273-4

Thébault E, Hulot G, Langlais B, Vigneron P (2021) A spherical harmonic model of Earth's lithospheric magnetic field up to degree 1050. Geophys Res Lett 48:e2021GL095147. https://doi.org/10.1029/2021GL095147

Töffner-Clausen L, Lesur V, Olsen N, Finlay CC (2016) In-flight scalar calibration and characterisation of the Swarm magnetometry package. Earth Planets Space 68:129. https://doi.org/10.1186/s40623-016-0501-6

Vigneron P, Hulot G, Olsen N, Léger J-M, Jager T, Brocco L, Sirol O, Coisson P, Lalanne X, Chulliat A, Bertrand F, Boness A, Fratter I (2015) A 2015
International Geomagnetic Reference Field (IGRF) candidate model based on Swarm's experimental absolute magnetometer vector mode data. Earth Planets Space 67:95. https://doi.org/10.1186/s40623-015-0265-4

Vigneron P, Hulot G, Léger J-M, Jager T (2021) Using improved Swarm's experimental absolute vector mode data to produce a candidate Definitive Geomagnetic Reference Field (DGRF) 2015.0 model. Earth Planets Space 73:197. https://doi.org/10.1186/s40623-021-01529-7

Vigneron P (2019) Mesures vectorielles expérimentales des instruments ASM de la mission SWARM: du commissioning à la production de modèles de champs géomagnétiques, Ph.D. Thesis, Université de Paris.

Yang Y, Zhou B, Hulot G, Olsen N, Wu Y, Xiong C, Stolle C, Zeren Z, Huang J, Zhu X, Pollinger A, Cheng B, Magnes W, Zhao X, Shen X (2021) CSES high precision magnetometer data products and example study of an intense geomagnetic storm. J Geophys Res Space Phys 126:e2020JA028026. https://doi.org/10.1029/2020JA028026

Zhang K (2023) A novel geomagnetic satellite constellation: science and applications. Earth Planet Phys 7(1):4–21. https://doi.org/10.26464/epp2023019

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