# Characterizing the magnetic signal generated in the magnetosphere from 1996 to 2024 using ground geomagnetic data

Yalei Shi, Vincent Lesur, Erwan Thébault

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Yalei Shi, Vincent Lesur, Erwan Thébault. Characterizing the magnetic signal generated in the magnetosphere from 1996 to 2024 using ground geomagnetic data. Earth Planets and Space, 2026, 10.1186/s40623-025-02363-x. hal-05465209

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# Characterizing the magnetic signal generated in the magnetosphere from 1996 to 2024 using ground geomagnetic data

Received: 02 Aug 2025

Accepted: 25 Dec 2025

Published online: 17 January 2026

Cite this article as: Shi, Y., Lesur, V., Thébault, E. et al. Characterizing the magnetic signal generated in the magnetosphere from 1996 to 2024 using ground geomagnetic data. *Earth Planets Space* (2026). https://doi.org/10.1186/s4062 3-025-02363-x

Yalei Shi, Vincent Lesur & Erwan Thébault

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# Characterizing the magnetic signal generated in the magnetosphere from 1996 to 2024 using ground geomagnetic data

Yalei Shi*, Université Paris Cité, Institut de Physique du Globe de Paris, CNRS, 75005

Paris, France, shi@ipgp.fr

Vincent Lesur, Université Paris Cité, Institut de Physique du Globe de Paris, CNRS, 75005

Paris, France

Erwan Thébault, Université Clermont Auvergne, Laboratoire Magmas et Volcans, OPGC,

IRD, CNRS, Aubière, France

<figure><img src="image_2.png" /></figure>
**Abstract**

The geomagnetic field, as observed at the Earth's surface or LEO satellite altitudes (between 300 km and 800 km), is the combination of signals generated by various internal and external sources. The internal sources are mainly associated to the liquid outer core flow, magnetized rocks in the lithosphere and induced electric currents in the crust and mantle. External sources are electric currents flowing in the ionosphere and magnetosphere. We focus on the contributions from the magnetospheric fields and describe a modeling approach in Spherical Harmonics (SH) based on magnetic observatory vector field measurements. The aim of this study is to model the magnetospheric field contributions observed during geomagnetically quiet time up to SH degree 6, with a one-hour temporal resolution for the period covering years 1996.0 to 2024.8. The adopted modeling approach is based on the Kalman filter and the correlation-based technique, which leads to series of hourly snapshot models together with robust error estimates. The series of models in time compare well with the global magnetospheric Ring Current index (RC). We observed and described various magnetospheric field structures, including local time asymmetries and contributions from ring and magnetotail currents. We also examined annual, semi-annual, monthly and daily variations in magnetospheric field Gauss coefficients in the Fourier domain.

**Keywords**

Magnetosphere, Geomagnetic field, Kalman filter, Ground-based observatory

**1 Introduction**

The geomagnetic field is a complex and dynamic electromagnetic phenomenon generated by sources both internal and external to the Earth (Hulot et al., 2010). Internally, the field originates primarily from geodynamo processes in Earth's liquid outer core, magnetized rocks in the lithosphere, and induced currents generated in Earth's conductive mantle and crust. Externally, the field is generated by electric currents in the ionosphere and magnetosphere. Beyond the Earth's magnetopause, the Interplanetary Magnetic Field (IMF) becomes the dominant magnetic force (Kavanagh Jr et al., 1970). Within the magnetopause, external sources vary across a wide range of temporal and spatial scales. Rapid temporal variations in
the near-Earth environment are largely driven by the interaction between Earth's magnetic field and the interplanetary magnetic field (IMF). The study of magnetic phenomena resulting from this coupling forms the foundation of space weather.

At a distance from the magnetic sources, a possible way to separate these different internal and external field contributions is to rely on magnetic potential theory and use Spherical Harmonics (SH). However, this separation remains challenging as it requires measurements with homogeneous temporal and spatial data coverage (C. Finlay et al., 2017). Magnetic field measurements are available from different platforms and at different altitudes, from the surface to the altitudes of the Low Earth Orbiting satellites. Dedicated magnetic field satellite missions provide measurements with a global homogeneous spatial coverage but suffer from a coarse spatio-temporal resolution. This limits the ability to separate with accuracy external magnetic fields that vary typically within a few minutes. As a consequence, in most geomagnetic field models derived from satellite data (C. Finlay et al., 2017), the time-varying magnetospheric contributions are generally poorly described. In contrast to satellite surveys, the ground geomagnetic observatory network provides continuous worldwide datasets with high temporal resolution. However, this dataset is more difficult to deal with because magnetic observatories are isolated, located on continents or islands (often of volcanic origin), and therefore are more sensitive to local electromagnetic field phenomena. These combined difficulties require careful data and meta-data analyses and statistical treatment to mitigate their effect in terms of biases in final global models. Yet, magnetic observatory networks are in principle well suited to monitor and model the time variations of large-spatial-scale magnetospheric signals and to highlight transient external magnetic field phenomena. Better describing these contributions would not only be beneficial to space weather activities but also to constraining other source field contributions such as, for example, the Earth's crustal magnetic field contributions aligned along the satellite orbits (Thébault et al., 2012; Lesur et al., 2013; Thébault et al., 2021).

The primary electric current systems in Earth's magnetosphere are the Chapman-Ferraro dayside magnetopause current, the magnetotail currents with closure via return currents on the nightside magnetopause, the Birkeland currents linked to the magnetosphere and ionosphere coupling, and the equatorial ring current, which contributes the most to the magnetic field in the near-Earth magnetic environment (Ganushkina et al., 2018). Magnetic fields in the magnetosphere can be studied by various methods.
Some are physics-based methods that rely on the principles of plasma dynamics and electromagnetic theory, making use of fundamental equations and established boundary conditions to numerically predict the Earth-Sun environment. These physics-driven models provide insight into the expected dynamics of the magnetosphere (e.g. Ogino, 1986; Raeder, 2003; Wang et al., 2013) but may disagree with direct magnetic field observations. In contrast, other methods rely primarily on magnetic field observations either at high altitudes (e.g. Cluster satellite mission, Escoubet et al. (1997)), low orbiting satellites (e.g. Swarm ESA mission, Friis-Christensen et al. (2008)), or at the ground level at magnetic observatories (e.g. INTERMAGNET network, Kerridge (2001)). These data-driven models are more phenomenological and may suffer from contamination from other magnetic sources. They require careful data pre-processing and model post-processing analyses for their assessment. They often have a low spatial resolution as a result of the limited global spatio-temporal coverage of the available data. In the context of the Swarm satellite mission (Olsen et al., 2013), one of such types of models is produced in SH and in near real time (Hamilton, 2013). It describes a SH degree 1 external magnetospheric field together with the Earth's induced counterpart, but is symmetric in longitude (SH order $m = 0$). More recently, Fillion et al. (2023) built a dedicated magnetospheric model from a joint analysis of the ground observatory with a one-hour time resolution and expanded up to SH degree 2. However, as described by the authors, the model is not regularized and suffers from some contamination by the ionospheric field, particularly during geomagnetically quiet times. Magnetospheric geomagnetic disturbances can also be monitored thanks to other proxies computed from ground-based magnetic field measurements such as the Disturbance-Storm-Time (Dst) index (Sugiura & Hendricks, 1967), the Ring Current (RC) index (Olsen et al., 2014), the SYM-H index (Iyemori et al., 2010), the SMR index (Newell & Gjerloev, 2012) or the VMD index (Thomson & Lesur, 2007). These indices are particularly useful to monitor the fast variations (one minute to one hour time resolution) of the equatorial symmetric ring current system, but their general baselines may differ. Finally, hybrid models can be constructed relying mostly on magnetic field measurements under constraints derived from plasma physics. One of such models, proposed by Tsyganenko (2013), is a well-known semi-empirical model with good spatio-temporal resolution, driven both by the theory of current system dynamics and by spacecraft data. The Tsyganenko model series, as well as other models reviewed in detail in Tsyganenko et al. (2021), focus on the description of the distant magnetospheric field, from several Earth's radii up to the magnetopause. Their agreement with ground-based magnetic field measurements in near-Earth magnetic environment is statistically acceptable during active geomag-
netic times but less during quiet times compared to independent magnetic field observations (Castillo et al., 2017).

In this study, we focus on the data-driven approach seeking the best compromise between spatio-temporal resolution and minimization of other source field contamination. In Section 2 we describe the magnetic observatory measurements we collected, their selection criteria, and the pre-processing and corrections we applied. We also present the parameterization of each magnetic field source that is co-estimated. In Section 3 we detail the steps of the modeling process. In particular, we summarize the principles of the Kalman filter approach (Kalman et al., 1960) combined with a correlation-based technique (Holschneider et al., 2016) that we use to incorporate realistic priori information on the data errors and model variances. Section 4 describes the model and the different criteria we used to assess it. In Section 5, the dynamics of the magnetosphere as predicted by the hourly magnetospheric field model is discussed and we conclude in Section 6. The Abbreviations section lists the full names of all the acronyms and abbreviations used in this paper.

## 2 Magnetic field measurements

### 2.1 Data Selection

INTERMAGNET is a worldwide global network of cooperating magnetic observatories that measure in real-time the vector magnetic field and its intensity (Kerridge, 2001). The measurements follow standard specifications and data format to ease data exchange and the production of geomagnetic auxiliary products in near-real time, such as magnetic field indices, for example. The dissemination of data in near-real time is ensured thanks to Geomagnetic Information Nodes (GINs) and a database is built upon definitive or quasi-definitive measurements. The INTERMAGNET specifications require a data noise variance not exceeding $1 \text{ nT}^2$ for the northward, eastward, and downward magnetic field vector components, respectively. In this work, we use hourly definitive measurements, corrected for baseline variations, and cleaned from identified spikes and erroneous data (Macmillan & Olsen, 2013). The vector and scalar database spans the period from 1996.0 (1st January 1996) to 2024.8 (30th October 2024), covering more than two solar cycles.

The observatory data record the geomagnetic signal originating from all geomagnetic field sources. In
order to minimize the contributions of fast varying fields and strong induced currents, we select the data corresponding to geomagnetically quiet times relying on the Dst index and keeping data corresponding to $|Dst|< 30$ nT. Then, to better highlight the contributions from the magnetosphere, we correct the data for core field contributions using the MCM and Kalmag core field models (Ropp & Lesur, 2023; Baerenzung et al., 2022) that are updated in the framework of the Swarm mission product services and cover the full time range (see the Data availability statement).

From 1996.0 to 2024.8, a total of 218 observatories provided data, although not all continuously over time. The geographic distribution of these observatories is shown in the left panel of Figure 1. Synthetic numerical simulations carried out in SH (not shown) indicated that this global spatial coverage allows in principle to separate the internal and the external fields up to SH degree 6. One of the main challenges with the INTERMAGNET database is to deal with the varying geographic coverage in time that may introduce artificial time variations (Figure 1, right panel). In particular, we see that the number of observatories meeting the INTERMAGNET standards decreases rapidly after epoch 2021.

## 2.2 Parameterization

After core field corrections, the selected geomagnetic field vector measurements $\mathbf{B}(r, \theta, \phi, t)$ are considered to be mainly the sum of the contributions from the magnetospheric field $\mathbf{B}^{\text{magneto}}(r, \theta, \phi, t)$ and its induced response in the mantle $\mathbf{B}^{\text{induced}}(r, \theta, \phi, t)$, from crustal field $\mathbf{B}^{\text{litho}}(r, \theta, \phi)$, and from ionospheric perturbations $\mathbf{B}^{\text{iono}}(r, \theta, \phi, t)$, written as

$$ \mathbf{B}(r, \theta, \phi, t) = \mathbf{B}^{\text{magneto}}(r, \theta, \phi, t) + \mathbf{B}^{\text{induced}}(r, \theta, \phi, t) + \mathbf{B}^{\text{litho}}(r, \theta, \phi) + \mathbf{B}^{\text{iono}}(r, \theta, \phi, t), \quad (1) $$

The magnetic contributions from the magnetospheric field and its induced counterparts are modeled through the gradient $\nabla_{r,\theta,\phi}$ of potentials described using SH and the Gauss coefficients $q_l^m$ and $i_l^m$, respectively. According to previous studies (e.g. Maus & Lühr, 2005), the magnetic signal of the magnetospheric current system (especially the ring current) is better represented in the Solar Magnetic coordinate system (SM) in the near-Earth magnetic environment. We use here two systems of coordinates for the parameterization. The contribution of the primary magnetospheric field is expressed in the SM
<figure><img src="image_3.png" /><figcaption>Figure 1. : Left-panel: the distribution of the ground geomagnetic observatory positions over a 29-year period. Right-panel: the number of observatories with vector data available at one-hour timestep between 1996.0 and 2024.8.</figcaption></figure>
coordinate system, with $\theta_{SM}$ and $\phi_{SM}$ the co-latitude and the longitude respectively, the associated Gauss coefficients are denoted $^{SM}q_{\ell}^{m}$. The response induced by the primary magnetospheric field in the Earth's mantle is expressed in the Earth-centered Earth-fixed coordinate system (ECEF), with $\theta$ and $\phi$ the co-latitude and longitude respectively, the associated Gauss coefficients are denoted $i_{\ell}^{m}$. In both systems of coordinates $r$ is the spherical radius. Then, the magnetospheric field and internally induced field can be represented as:

$$ \mathbf{B}^{\text{magneto}}(r, \theta, \phi, t) = -a \nabla_{r,\theta,\phi} \sum_{\ell,m}^{L} {}^{SM}q_{\ell}^{m}(t) \left(\frac{r}{a}\right)^{\ell} Y_{\ell}^{m}(\theta_{SM}, \phi_{SM}), \quad (2) $$

$$ \mathbf{B}^{\text{induced}}(r, \theta, \phi, t) = -a \nabla_{r,\theta,\phi} \sum_{\ell,m}^{L} i_{\ell}^{m}(t) \left(\frac{a}{r}\right)^{\ell+1} Y_{\ell}^{m}(\theta, \phi), \quad (3) $$

where $a = 6371.2$ km is the usual Earth's spherical reference radius and $\sum_{\ell,m}^{L}$ stands for the double sum $\sum_{\ell}^{L} \sum_{m=-\ell}$, with $L$ the maximum SH degree, $m$ the order. $Y_{\ell}^{m}$ are the Schmidt semi-normalized SH functions.

The lithospheric field varies significantly at local scales, and no SH representation is currently available with a spatial resolution better than 30 km that would have allowed a deterministic correction for these contributions (e.g. Thébault et al., 2021). The lithospheric field is therefore modeled as a local bias for each observatory, that is parameterized by the vector $^{L}\mathbf{O}_{i}$, and the Kroenecker $\delta$ distribution. They are different biases ($^{L}\mathbf{O}_{r}^{i}$, $^{L}\mathbf{O}_{\theta}^{i}$, $^{L}\mathbf{O}_{\phi}^{i}$) for each vector component at each observatory denoted by the subscript $i$, with $r_{i}$, $\theta_{i}$, and $\phi_{i}$, the spherical coordinates of the observatory. It is assumed that biases are not correlated in space and are nearly static in time. Thanks to the long series of observatory measurements, some statistics are available for each of these contributions and are discussed in the next section. The lithospheric field can be represented as:

$$ \mathbf{B}^{\text{litho}}(r, \theta, \phi) = \sum_{i \in N_{\text{obs}}} \delta(r - r_i, \theta - \theta_i, \phi - \phi_i)^{L} \mathbf{O}_{i}, \quad (4) $$

where $N_{\text{obs}}$ is the number of used observatories.

A similar parameterization is set up to account for ionospheric fields, since we use both day-side and night-side data. One possible approach to account for this contribution could have been to co-estimate a global SH representation. However, this treatment of the ionospheric fields requires a SH expansion
up to a SH degree between $L = 45$ and $L = 60$ (Chulliat et al., 2013; Sabaka et al., 2015), and thus a large number of parameters describing the primary and induced ionospheric fields must be resolved numerically, which is not achievable with the number of available magnetic observatory data. Therefore, we follow an approach whereby the ionospheric fields are considered as perturbations in the observed measurements at each observatory. These ionospheric vector perturbations defined by the vector $^iO_i$ are also assumed to be different for each observatory $i$ and for each component $^iO_r$, $^iO_\theta$, and $^iO_\phi$. They are represented as:

$$ \mathbf{B}^{\text{iono}}(r, \theta, \phi, t) = \sum_{i \in N_{\text{obs}}} \delta(r - r_i, \theta - \theta_i, \phi - \phi_i)^{\text{i}} \mathbf{O}_i(t). \quad (5) $$

They depend on the local time, and to guide this parameterization we rely on the TIEGCM model (Maute, 2017) and the solar radio flux index F10.7 using an approach presented in detail in the next section.

Accordingly, the geomagnetic field vector measurements can be represented as the sum of these contributions as:

$$ \begin{aligned} \mathbf{B}(r, \theta, \phi, t) = & -a \nabla_{r,\theta,\phi} \sum_{\ell,m}^{SM} q_{\ell}^{m}(t) \left(\frac{r}{a}\right)^{\ell} Y_{\ell}^{m}(\theta_{SM}, \phi_{SM}) \\ & -a \nabla_{r,\theta,\phi} \sum_{\ell,m}^{L} i_{\ell}^{m}(t) \left(\frac{a}{r}\right)^{\ell+1} Y_{\ell}^{m}(\theta, \phi) \\ & + \sum_{i \in N_{\text{obs}}} \delta(r - r_i, \theta - \theta_i, \phi - \phi_i)^{L} \mathbf{O}_i \\ & + \sum_{i \in N_{\text{obs}}} \delta(r - r_i, \theta - \theta_i, \phi - \phi_i)^{i} \mathbf{O}_i(t), \end{aligned} \quad (6) $$

As outlined in Section 2.1, as long as only large-scale SHs are investigated, the internally induced field can be well separated directly in SH from the magnetospheric field. This is not true when considering local crustal biases and smaller scales ionospheric fields. With the formalism described by Eq. (6), the separation of the magnetospheric primary and induced fields from other constituents is achieved by the priors that describe statistically the crustal and ionospheric fields (see next section).

### 3 Modeling process

We rely on a Kalman filter approach whose principles have proven their efficiency for geomagnetic field modeling, whether for Earth's core field (Fournier et al., 2015; Lesur et al., 2018; Baerenzung et al., 2018;
Barrois et al., 2018; Ropp et al., 2020), for lithospheric magnetic fields (Baerenzung et al., 2022), or for ionospheric magnetic field forecasts (Durazo et al., 2017).

We apply the Kalman filter approach to estimate the normal distributions of models that describe different contributions to magnetic observations in observatories (derived from Eq. (6)). The temporal evolution is given by a series of snap-shot normal distributions at one hour time steps. A normal distribution $N(\mu = \mathbf{m}, \sigma^2 = \mathbf{C}_m)$ is defined by an hourly mean model, denoted by $\mathbf{m}$, and a corresponding covariance matrix, denoted by $\mathbf{C}_m$. The hourly mean model at each time step ($\mathbf{m}_k$) consists of external and internal SH Gauss coefficients ($^{SM}\mathbf{q}_\ell^m, i_\ell^m, \ell=1,...,6$), the lithospheric contribution ($^L\mathbf{O}_i, i \in N_{obs}$), and the ionospheric perturbations ($^I\mathbf{O}_i, i \in N_{obs}$). Each hourly model ($\mathbf{m}_k$) has a total number of 1404 parameters (see the first and last lines in Table 1), and each covariance matrix ($\mathbf{C}_{m,k}$) corresponding to the hourly model ($\mathbf{m}_k$) is square with dimension 1404×1404. The series of 254, 214 snap-shot distributions spans a period of 29 years from 1996 to 2024 during geomagnetically quiet times.

We compute the hourly model distributions characterized by the mean model $\mathbf{m}_k = \delta\mathbf{m}_k + \mathbf{m}_0$ and the covariance $\mathbf{C}_{m,k}$ for each time step of one hour, where the a priori mean model $\mathbf{m}_0$ and the a priori covariance matrix $\mathbf{C}_0$ are used as the initial state of the process. $\delta\mathbf{m}_k$ is initially set to 0 ($\delta\mathbf{m}_0 = 0$). The Kalman filter, starting from the initial state, operates through a three-step process to produce a time series. It first starts with the analysis step during which the model is adjusted by fitting INTERMAGNET observatory data, based on the Bayesian inversion scheme. Then, the prediction step projects the current model forward to the next time step using a prediction operator. Finally, a smoothing step is applied a posteriori on the series of models, based on the assumption of a Gaussian correlation over time. Each of these steps is detailed in the following subsections.

## 3.1 A priori information

Each model $\mathbf{m}_k$ involved in Eq. (6) is expected to fluctuate around an initial value $\mathbf{m}_0$ within the allowed range defined by the correlation covariance matrix $\mathbf{C}_{m,k}$ initially set to $\mathbf{C}_0$.

The a priori mean model $\mathbf{m}_0$ is made of Gauss coefficient values (refer to the second row of Table 1) that are null for the magnetospheric contribution and for the induced fields, with the exception of the symmetric axial dipole coefficients of the external magnetospheric field contribution. This a priori value
of external Gauss coefficients of SH degree 1 and order 0 is defined as $q_1^0 = 7.57$ nT in the SM coordinates. It corresponds to a steady southward quiet-time magnetospheric field, as determined through the study of Maus & Lühr (2005). As described by Maus & Lühr (2005), this field is the combined contribution of fields generated from both the ring current and the Chapman-Ferraro currents during geomagnetically quiet times. The a priori mean model $m_0$ also includes the statistical average of crustal biases and ionospheric perturbations for each component at each observatory. The details of these values and their corresponding covariance matrix are discussed below.

The initial covariance matrix $C_0$ (see the third row of Table 1) is an a priori block-diagonal covariance matrix corresponding to the initial model $m_0$. The a priori block covariance matrix corresponding to the magnetospheric and induced fields are diagonal, with variances $v_0^{ext}(\ell)$ and $v_0^{int}(\ell)$ for the magnetospheric field and its induced counterpart, respectively. These variances are defined by the type of a priori information described in Holschneider et al. (2016) based on the expected energy spectrum of each contribution. Each energy spectrum is parameterized by a scaling factor $S_{ns}^e$ (resp. $S_{ns}^i$) and a radius $R_{ns}^e$ (resp. $R_{ns}^i$) at which the respective source energy spectra are assumed flat. The derived variances are:

$$v_0^{ext}(\ell) = S_{ns}^e \left(\frac{a}{R_{ns}^e}\right)^{2\ell}, \qquad (7)$$

$$v_0^{int}(\ell) = S_{ns}^i \left(\frac{R_{ns}^i}{a}\right)^{2\ell+2}. \qquad (8)$$

The scaling factor $S_{ns}^e$ and the radius $R_{ns}^e$ for the magnetospheric field are estimated from its quiet-time steady energy spectrum. The latter has been derived from the quiet-time steady $q_\ell^m$ magnetospheric values in both the SM and Geocentric Solar Magnetic (GSM) coordinates, as given by Maus & Lühr (2005). In this paper, the authors show that these $q_\ell^m$ describe contributions from the ring currents, the Chapman-Ferraro currents, and the magneto-tail currents. Regarding the induced field power spectrum (Eq. (8)), the scaling factor $S_{ns}^i$ and the radius $R_{ns}^i$ are defined according to the a priori relationship between the primary magnetospheric field and its induced part through the magnetic indices $I_{st}$ (internal part of the Dst index) and $E_{st}$ (external part of the Dst index), where $I_{st} = 0.32E_{st}$ (Maus & Weidelt, 2004). Combining these a priori information provides the relation between both spectra:

$$v_0^{int}(\ell) = (0.32^2)v_0^{ext}(\ell). \qquad (9)$$
We now turn to the a priori mean model values contained in $\mathbf{m}_0$ that describe the statistical average of the crustal biases and ionospheric perturbations for each component at each observatory. The a priori constant value of crustal biases $^L\mathbf{O}_i$ ($i \in N_{obs}$) and its corresponding initial covariance matrix $^L\mathbf{C}_0^i$ for the 3 components at each observatory are, respectively, defined as the statistical mean and the statistical variance of the full time series of the data used, with the core and external field contributions subtracted. The subtracted core contributions are given by the MCM model (Ropp & Lesur, 2023) and the Kalmag model (Baerenzung et al., 2022). The subtracted external contributions are given by the Dst index field along the axial dipole direction.

The initial ionospheric perturbation parameters $^I\mathbf{O}_i$ ($i \in N_{obs}$), are derived from the TIEGCM model. The time series of this model estimated for each hour, over the year 2009 of minimum solar activity level and over the year 2002 of maximum solar activity level, have been given by Maute (2017). The statistical estimation of 24 mean values can be derived, one for each hour within a day, at each observatory site for each component, from the time series of this model for the years 2002 and 2009, respectively. Similarly, correlations among all observatory components can be estimated at each hour, leading to a full a priori covariance matrix ($^I\mathbf{C}_0^i$) dependent on local time. The so-derived ionospheric perturbations exhibit a direct correlation with the Sun's activity. As a result, the a priori perturbation values for each component at each observatory site should depend on local time and solar activity. The solar activity levels are characterized by the solar F10.7 index. It follows that the a priori distribution of the ionospheric parameters can be estimated for a given year using linear interpolation based on the F10.7 annual mean value in comparison to the F10.7 levels for which the TIEGCM is available – i.e. F10.7 annual mean value for 2002 and 2009.

## 3.2 Analysis step

The analysis step adjusts the hourly model $\mathbf{m}_k = \delta\mathbf{m}_k + \mathbf{m}_0$ by fitting the observatory data in each hourly time step ($t_k$) based on the least squares (LS) method. The available data for the 218 observatories in the $k^{th}$ one-hour time step ($\mathbf{B}(r_i, \theta_i, \phi_i, t_k)$ with $i \in N_{obs}$) are gathered in a vector $\mathbf{d}_k$. The model parameters $\delta\mathbf{m}_k + \mathbf{m}_0$ and the data are connected by the linear operator $\mathbf{A}$

$$ \mathbf{d}_k = \mathbf{A}(\delta\mathbf{m}_k + \mathbf{m}_0) + \mathbf{e}, \qquad (10) $$
<table><caption>Table 1. : The parameters of modeled sources. From top to bottom: the source, coefficient notation used in Eq. (6), a priori mean models, corresponding a priori covariance, time scales, inverse problem matrix in "Analysis step", minimum ($L_i$) and maximum ($L_a$) harmonic degrees, and the total number of parameters. $ℓ$ is SH degree, $m$ is SH order, $Y_ℓ^m$ is Schmidt semi-normalized SH function.</caption><thead><tr><th></th><th>magnetosphere</th><th>induced field</th><th>crustal biases</th><th>Ionosphere</th></tr></thead><tbody><tr><td>Coef.</td><td><sup>SM</sup><i>q</i><sub>ℓ</sub><sup>m</sup></td><td><i>i</i><sub>ℓ</sub><sup>m</sup></td><td><sup>L</sup><b>O</b><sub>i</sub>=<sup>L</sup>(<i>O</i><sub>i,r</sub>, <i>O</i><sub>i,θ</sub>, <i>O</i><sub>i,φ</sub>)</td><td><sup>I</sup><b>O</b><sub>i</sub>=<sup>I</sup>(<i>O</i><sub>i,r</sub>, <i>O</i><sub>i,θ</sub>, <i>O</i><sub>i,φ</sub>)</td></tr><tr><td>A priori<br>Mean<br>(<b>m</b><sub>0</sub>)</td><td><i>q</i><sub>l=1</sub><sup>m</sup> ≠ 0, <i>q</i><sub>l>1</sub><sup>m</sup> = 0</td><td>0</td><td>From data series</td><td>From<br>model<br>TIEGCM</td></tr><tr><td>A priori<br>Covariance<br>(<b>C</b><sub>0</sub>)</td><td><i>R</i><sub>ns</sub><sup>e</sup>=3.5.10<sup>4</sup>km<br><i>S</i><sub>ns</sub><sup>e</sup>=0.7.10<sup>4</sup> nT<sup>2</sup></td><td><i>R</i><sub>ns</sub><sup>i</sup>=1.2.10<sup>3</sup>km<br><i>S</i><sub>ns</sub><sup>i</sup>=5.10<sup>2</sup> nT<sup>2</sup></td><td>From data series</td><td>From<br>model<br>TIEGCM</td></tr><tr><td>Time<br>scale</td><td><i>τ</i><sup>e</sup> = 10 hour</td><td><i>τ</i><sup>i</sup> = 10 hour</td><td><i>τ</i><sup>L</sup> = 10<sup>6</sup> year</td><td><i>τ</i><sup>iono</sup> = 24 hour</td></tr><tr><td>Source<br>reference</td><td>Maus &amp; Lihr<br>(2005), Hnat (2019)</td><td>Maus &amp; Weidelt<br>(2004)</td><td>Ropp et al. (2020)</td><td>Maute (2017)</td></tr><tr><td>Matrix <b>A</b></td><td>∇<sub>r,θ,φ</sub><i>a</i>(<sup><i>r</i></sup>/<sub><i>a</i></sub>)<sup><i>ℓ</i></sup><i>Y</i><sub>ℓ</sub><sup>m</sup></td><td>∇<sub>r,θ,φ</sub><i>a</i>(<sup><i>a</i></sup>/<sub><i>r</i></sub>)<sup><i>ℓ</i>+1</sup><i>Y</i><sub>ℓ</sub><sup>m</sup></td><td>1</td><td>1</td></tr><tr><td><i>L</i><sub>i</sub>, <i>L</i><sub>a</sub></td><td>1, 6</td><td>1, 6</td><td>Non SH</td><td>Non SH</td></tr><tr><td>Nb.</td><td>48</td><td>48</td><td>3×218</td><td>3×218</td></tr></tbody></table>

where **e** is a vector of uncorrelated random data errors. The elements of **A** are derived from Eq. (6) (see also the 6th row in Tab. 1). The least squares solution of each hourly model $\delta\mathbf{m}_k + \mathbf{m}_0$ and its covariance matrix $\mathbf{C}_{m,k}$ are given by

$$ \delta\mathbf{m}_k = \delta\tilde{\mathbf{m}}_k + \tilde{\mathbf{C}}_{m,k}\mathbf{A}^t(\mathbf{A}\tilde{\mathbf{C}}_{m,k}\mathbf{A}^t + \mathbf{C}_e)^{-1}(\mathbf{d}_k - \mathbf{A}(\delta\tilde{\mathbf{m}}_k + \mathbf{m}_0)), \quad (11) $$

$$ \mathbf{C}_{m,k} = \tilde{\mathbf{C}}_{m,k} - \tilde{\mathbf{C}}_{m,k}\mathbf{A}^t(\mathbf{A}\tilde{\mathbf{C}}_{m,k}\mathbf{A}^t + \mathbf{C}_e)^{-1}\mathbf{A}\tilde{\mathbf{C}}_{m,k}, \quad (12) $$

where $\mathbf{C}_e$ is the covariance matrix of the data errors. $\delta\tilde{\mathbf{m}}_k$ and $\tilde{\mathbf{C}}_{m,k}$ are the a priori model and covariance matrix for $\delta\mathbf{m}_k$ respectively. To handle possible outliers in the distribution of data errors, the solution is derived through an iterative LS method, in three iterations, using Huber weights. In the first step $\delta\tilde{\mathbf{m}}_1 = \delta\mathbf{m}_0 = 0$ and $\tilde{\mathbf{C}}_{m,1} = \mathbf{C}_0$.

### 3.3 Prediction step

The prediction step projects the hourly model $\delta\mathbf{m}_k$ obtained in the analysis step Eq. (11), to the next time step ($\delta\tilde{\mathbf{m}}_{k+1}$) assuming the model follows an auto-regressive process of order 1 (AR1). The predicted
hourly model $\delta\tilde{\mathbf{m}}_{k+1}$ and its corresponding covariance matrix $\tilde{\mathbf{C}}_{m,k+1}$ are defined by the following:

$$ \delta\tilde{\mathbf{m}}_{k+1} = \mathbf{P}\delta\mathbf{m}_k + \mathbf{w}, \qquad (13) $$

$$ \tilde{\mathbf{C}}_{m,k+1} = \mathbf{P}\mathbf{C}_{m,k}\mathbf{P}^t + \mathbf{C}_w, \qquad (14) $$

where $\mathbf{P}$ is the prediction operator and the prediction error $\mathbf{w}$ is a multivariate random variable with zero mean and diagonal covariance matrix $\mathbf{C}_w$. The AR1 operator $\mathbf{P}$ is a diagonal matrix described by a prediction variable $\alpha$, in such a way that

$$ \alpha = \begin{cases} e^{-\frac{\Delta t}{\tau}}, & \tau > \Delta t, \\ 0, & \tau < \Delta t. \end{cases} \qquad (15) $$

The prediction time step $\Delta t$ is one hour, and the time scale $\tau$ depends on the different geomagnetic sources. Values for each source field are given in Table 1. When the time scale is much smaller than the prediction time step, the prediction error is the largest and corresponds to the background state. Otherwise, $\mathbf{C}_w$ writes as $\mathbf{C}_w = \mathbf{C}_0 - \mathbf{P}\mathbf{C}_0\mathbf{P}^t$.

## 3.4 Smoothing step

At the end of the entire iterative process over the full time range, the a posteriori smoothing step is applied to correct the hourly model backward in time starting from the last derived model distribution.

The smoothed model at step $k$, $\delta\mathbf{m}_k^s$, can be seen as the solution of the linear inverse problem $\delta\mathbf{m}_{k+1}^s = \mathbf{P}\delta\mathbf{m}_k^s + \mathbf{w}$, where the model $\delta\mathbf{m}_{k+1}^s$ is a "data" vector with covariance $\mathbf{C}_{m,k+1}^s$. This leads to

$$ \delta\mathbf{m}_k^s = \delta\mathbf{m}_k + \mathbf{G}_k(\delta\mathbf{m}_{k+1}^s - \mathbf{P}\delta\mathbf{m}_k), \qquad (16) $$

$$ \mathbf{C}_{m,k}^s = \mathbf{C}_{m,k} - \mathbf{G}_k(\mathbf{P}\mathbf{C}_{m,k}\mathbf{P}^t + \mathbf{C}_w - \mathbf{C}_{m,k+1}^s)\mathbf{G}_k^t, \qquad (17) $$

where the matrix $\mathbf{G}_k$ is defined as

$$ \mathbf{G}_k = \mathbf{C}_{m,k}\mathbf{P}^t(\mathbf{P}\mathbf{C}_{m,k}\mathbf{P}^t + \mathbf{C}_w)^{-1}. \qquad (18) $$
After this smoothing step, the series of models from 1996 to 2024.8, are in their final version ($\delta\mathbf{m}_k^s + \mathbf{m}_0$) and provided with robust error estimations ($C_{m,k}^s$). The modeled geomagnetic field for each time step is defined as $\mathbf{B}^{\text{model}}(r, \theta, \phi, t_k) = \mathbf{A}(\delta\mathbf{m}_k^s + \mathbf{m}_0)$.

# 4 Results

We use three criteria to assess our hourly magnetospheric field model developed to SH degree 6. The first criterion is to look at the misfit between the geomagnetic modeled values ($\mathbf{B}^{\text{model}}$) and the observed data ($\mathbf{B}^{\text{obs}}$). The second investigates how much useful information was extracted from the observatory data to improve the a priori information. Finally, the third criterion is based on a comparison between the hourly model ($q_1^0, i_1^0$) estimations and geomagnetic indices such as the RC and Dst indices. The statistics of the modeled Gauss coefficients are given in the last sub-section.

## 4.1 Data misfit

We compute data misfits between the geomagnetic observations and the model estimations from 1st January 1996 to 30th October 2024 during geomagnetically quiet times. Figure 2 shows the average standard deviation ($\sigma$, in nT) of data misfits over the analyzed periods for each observatory used in this study. We notice that the average standard deviation of the misfit is less than 10 nT for observatories at mid- and low-latitudes. In contrast, larger values are observed at 32 high-latitude observatories, all of which are located outside the $\mp60^\circ$ quasi-dipole (QD) latitudinal band. At these positions, magnetic measurements could be strongly influenced by rapid and small-scale magnetic field variations produced by polar electrojets. This misfit dichotomy is quantified in Table 2 and shows, in particular, a biased mean toward negative values for the North component ($x=-B_\theta$) at high latitudes. In general, such a bias indicates that some magnetic field structures are poorly taken into account in the model parameterization (Eq. (6)). However, low- and mid-latitude observatory measurements, corresponding to the here so-called "Quiet Observatories", are globally well fitted by the model parameterization with mean bias in general less than 1 nT in absolute value. The typical standard deviations remain below 4.5 nT on average for all three components, with maximum values of 13.6 nT for the North component at AAE and 16.1 nT for the North component at HUA (see Table 2 and Appendix A Figure 19). Except for the polar areas affected by contributions from the auroral electrojet, the agreement between the model and the data
is satisfactory considering both local internal induction effects and the 1 nT data error specification of INTERMAGNET data.

<table><caption>Table 2. : Means and standard deviations of the residuals derived from all 218 observatories data from 1st January 1996 to 30th October 2024 during geomagnetically quiet times. Units are nT.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Perturbed Obs (|QD-Lat|≥60°)</th><th colspan="2">Quiet Obs (|QD-Lat|<60°)</th><th colspan="2">Full Obs</th></tr><tr><th>mean</th><th>std</th><th>mean</th><th>std</th><th>mean</th><th>std</th></tr></thead><tbody><tr><td>x component</td><td>-3.36</td><td>36.17</td><td>0.004</td><td>4.58</td><td>-0.78</td><td>18.23</td></tr><tr><td>y component</td><td>-0.12</td><td>20.28</td><td>0.12</td><td>4.18</td><td>0.06</td><td>10.54</td></tr><tr><td>z component</td><td>-0.12</td><td>29.28</td><td>-0.02</td><td>2.66</td><td>-0.04</td><td>14.48</td></tr></tbody></table>

To test our models against independent data, we use definitive geomagnetic data from the SOK observatory for year 2021, which were not used during the modeling process. The SOK observatory is located in Sop, Senegal (14.5°N, 16.5°W), North to the magnetic equator. We first correct the vector data for the geomagnetic core field contribution given by the MCM model (Ropp & Lesur, 2023), and select only night-time data from 23h LT to 5h LT to minimize particularly strong ionospheric disturbances near the magnetic equator. Then, we correct the measurements for the geomagnetic contributions of magnetospheric and induced fields, as estimated by our models. After this series of corrections, the SOK observatory vector residuals are primarily the sum of crustal fields and some noise, as summarized in Table 3. The mean residual corresponds to the crustal bias of the SOK observatory but the standard deviation, which contains data error, unknown geomagnetic field sources, and misfit residuals with our model does not exceed 5.1 nT for all magnetic conditions, which is within the range of values we obtain with the INTERMAGNET database (Table 2). In addition, Figure 3 provides an overview of the comparison over several days between the night-time residual field values from the SOK observatory (after removal of the core and lithospheric field contributions) and the magnetic contributions (magnetospheric and induced fields) estimated with our model, for the northward, eastward, and downward components, respectively. It shows that the model-estimated fields follow relatively well the observed magnetic signals at this independent observatory site.

## 4.2 Output variance estimates

We evaluate how much information our model extracts from the observatory data by comparing the a posteriori and the a priori variances. Figure 4 shows the percentage of reduction between the a priori and a posteriori variances of both the magnetospheric Gauss coefficients (left panel) and the internally
<figure><img src="image_4.png" /><figcaption>Figure 2. : During geomagnetically quiet times (1st Jan 1996 – 30th Oct 2024): Observatory sites with colors showing mean σ across north, east, and down components. QD: Quasi-Dipole coordinates.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3. : Comparison over several days between the night-time residual field values from the SOK observatory (after removal of the core and lithospheric contributions; cyan) and the magnetic contributions (magnetospheric and induced fields) estimated with our model (pink), for the northward, eastward, and downward components, respectively.</figcaption></figure>
<table><thead><tr><th></th><th>mean</th><th>std</th></tr></thead><tbody><tr><td>x component</td><td>52.96</td><td>4.55</td></tr><tr><td>y component</td><td>-21.86</td><td>4.23</td></tr><tr><td>z component</td><td>7.01</td><td>5.05</td></tr></tbody></table>

Table 3. : The means and the standard deviations of the residuals derived from the SOK observatory data over the year 2021 during quiet times. Estimates of the main contribution from the MCM model, along with the induced and external contributions from this study, have been removed from the raw hourly mean SOK observatory data. Units are nT.

induced Gauss coefficients (right panel), degree by degree. When the SH degrees are small, the a posteriori variances are significantly reduced. The a posteriori variances of the magnetospheric Gauss coefficients (resp. internal induced coefficients) are reduced by 99.58% (resp. 99.01%) for SH $\ell$ = 1, reduced by 94.81% (resp. 89.73%) for SH $\ell$ = 2, reduced by 63.17% (resp. 48.55%) for SH $\ell$ = 3 and reduced by 18.35% (resp. 5.46%) for SH $\ell$ = 4. We notice that at SH degrees 5 and 6, the a posteriori variances are almost identical to the a priori variances, and that the ratios are 2.24% (resp. 0.31%) for SH $\ell$ = 5, and 0.15% (resp. 0.01%) for SH $\ell$ = 6. This indicates that the model does not extract much information from the data for the SH degree larger than 4.

In addition, we looked at the variability of the a posteriori variance of our models within a day. Figure 5 (resp. Figure 6) shows the a posteriori variance of the first 4 SH degrees from 3 different days for magnetospheric coefficients (resp. internally induced coefficients). The first day is 21st October 2019 (left panel), which corresponds to the weakest solar activity (solar flux ratio = 63.4 sfu) covered by data from 126 operating observatories. The second day is 12th July 2000 (center panel), which corresponds to higher solar activity (solar flux ratio = 325.1 sfu) covered by 135 operating observatories. The last day is 17th March 2023 (right panel), which corresponds to the highest solar activity (solar flux ratio = 335 sfu) covered by only 79 operating observatories. The structure of the a posteriori variance is consistent across the three days. The a posteriori variance within a day varies between the maximum values (black lines) and the minimum values (grey lines), and the interval is filled with purple color. We observe that the posterior variance increases slightly with higher solar activity (left and center panels of Figure 5 and Figure 6) but that the estimates are relatively stable within a day by looking at the small difference between the maximum and the minimum of the a posteriori variances (purple-filled color range). However, the a posteriori variances clearly depend on the availability of magnetic observatories (center and right panels of Figure 5 and Figure 6) and the estimates vary significantly within a day, when there are few observatoires, as can be seen from the larger difference between maximum and minimum
of the a posteriori variances. The occurrences of maximum and minimum posterior variances of the magnetosphere Gauss coefficients also change during the days. As solar activity decreases (compare the center and left panels of Figure 5), the occurrences of maximum posterior variance shift from post-midnight to early morning. This change is less significant for the internally induced Gauss coefficients that always have a post-midnight posteriori variance maximum and a minimum at noon. This difference can be explained by the combined effects of an uneven repartition of observatories around the world and of the ionospheric disturbances.

## 4.3 Comparison with RC and Dst indices

The Dst is a scalar index describing the horizontal perturbation of the global magnetospheric magnetic field, derived from 4 ground geomagnetic observatories near the geomagnetic equator (Sugiura & Hen-dricks, 1967). The RC (Ring Current) index, as part of the series of CHAOS geomagnetic models, was defined as a southward-directed magnetospheric field at the dip equator (Olsen et al., 2014). The Dst and RC indices have been broken down into external (Est, RC<sup>e</sup>) and induced internal (Ist, RC<sup>i</sup>) indices by using a priori 1D conductivity models of the Earth's mantle (Maus & Weidelt, 2004; Olsen et al., 2005). We check the consistency of the hourly model with the RC and Dst indices. For this, we convert the external coefficient $q_1^0$ from the Solar Magnetic (SM) coordinate system to $\tilde{q}_1^0$ now expressed in the Earth-centered Earth-fixed (ECEF) coordinate system.

Figure 7 shows comparisons, from March to July 2009, between the negative estimated external dipole coefficient ($-\tilde{q}_1^0$, in pink) with the RC<sup>e</sup> index (in the top panel) and the comparisons between the estimated induced dipole coefficient ($-i_1^0$, in pink) with the RC<sup>i</sup> index (in the bottom panel). The 3σ variation ranges around the estimated $-\tilde{q}_1^0$ and $-i_1^0$ series are shown as shaded areas in purple color. The comparisons show a good agreement between the estimated internal and external dipole coefficients ($\tilde{q}_1^0, i_1^0$) with the RC indices (RC<sup>e</sup>, RC<sup>i</sup>). The offset between the $-i_1^0$ and the RC<sup>i</sup> time series in the bottom panel is explained by the average of our $-i_1^0$ model being close to 0, just as expected, while the RC<sup>i</sup> has a non-null mean value around 1.22 nT (see the average of Table 4). Figure 8 shows the correlation coefficients between the RC index (resp. Dst index) and the dipole coefficients, computed on a yearly basis over the study period. The red (resp. blue) dots are for external coefficients, and the pink (resp. cyan) dots are for internal coefficients. The correlations are generally satisfactory, particularly
<figure><img src="image_6.png" /><figcaption>Figure 4. : Percentage reduction between the posterior variance and the a priori variance. The left panel shows the variance ratio of magnetosphere Gauss Coefficients $q_l^m$, and the right panel shows the variance ratio of internal induced Gauss Coefficients $i_l^m$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5. : Hourly dependence of the a posteriori variance of modeled external coefficients on three dates: 21 Oct 2019 (weakest solar activity, 63.4 sfu), 12 Jul 2000 (second strongest, 325.1 sfu), and 17 Feb 2023 (strongest, 335 sfu). Black lines show daily maxima, grey lines minima, with the area between shaded purple. $sfu = 10^{-22} Wm^{-2}Hz^{-1}$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6. : The same as Figure 5, but for the modeled internally induced coefficients.</figcaption></figure>
with the RC index, but exhibit a downward tendency from 2021 to 2024 when the number of available
observatories decreases (see right panel of Figure 1). The correlations for internal coefficients (between
Ist and $-i_1^0$) are weaker during solar maximum years, such as 2002 and 2014.

For the whole quiet-time period spanning 1997 to 2024, during which the RC index is available, Table
4 shows the Root-Mean-Square Error (RMSE) and Correlation Coefficient (CC) values quantifying the
relationship between both the RC and Dst indices and the axial dipole coefficients derived from our
method. The CC values for both the external and internal parts are larger than 0.8. The RMSE values
are 7.43 nT and 2.47 nT for the Est and Ist index respectively, whereas it is lower for the $RC^e$ and $RC^i$
index, around 4.86 nT and 2.39 nT respectively. Our modeled dipole coefficients are in better agreement
with the RC than with the Dst indices. We recall that in our approach we did not use a priori information
on the mantle conductivity; nonetheless the correlation with the $RC^i$ and Ist indices is no less than 0.8.
It is important to note that we do not expect a perfect correlation between the magnetospheric model and
the RC or Dst indices. First, both indices may be subject to contamination from non-axially symmetric
magnetospheric and ionospheric fields. Second, their separation into internal and external components
relies on strong assumptions regarding the one-dimensional Earth's mantle conductivity. Third, the
contributions of Chapman-Ferraro current are not removed from our modeled dipole coefficient.

<table><caption>Table 4. : Comparison between the modeled dipole coefficients ($\tilde{q}_1^0$, $i_1^0$) and the magnetospheric activity indices (Dst, RC) from 1st January 1997 to 27th August 2024 during geomagnetically quiet times. The average value in the ECEF coordinate system is in nT. The Root Mean Square Error (RMSE) (in nT) and the Correlation Coefficient (CC) of the external part (resp. internal part) of the RC index or Dst index are related to minus $\tilde{q}_1^0$ (resp. minus $i_1^0$).</caption><thead><tr><th></th><th>Average</th><th>RMSE</th><th>CC</th></tr></thead><tbody><tr><td>-q̃<sup>0</sup><sub>1</sub></td><td>-6.17</td><td></td><td></td></tr><tr><td>RC<sup>e</sup></td><td>-7.44</td><td>4.86</td><td>0.88</td></tr><tr><td>Est</td><td>-5.38</td><td>7.43</td><td>0.8</td></tr><tr><td>-i<sup>0</sup><sub>1</sub></td><td>-0.22</td><td></td><td></td></tr><tr><td>RC<sup>i</sup></td><td>1.22</td><td>2.39</td><td>0.82</td></tr><tr><td>Ist</td><td>-1.72</td><td>2.47</td><td>0.79</td></tr></tbody></table>

## 4.4 Average and standard deviation of the SH model coefficients

The 29-year averages and the standard deviation of the estimated Gauss coefficients, up to the SH degree 6,
for the magnetospheric field in the SM coordinates and the internal induced field in the ECEF coordinates,
are presented in Table (5). The Gauss coefficient $q_1^0$ is the largest and positive value, indicating a general
<figure><img src="image_9.png" /><figcaption>Figure 7. : Comparison between the external part of the RC index ($RC^e$) and the external coefficient $\tilde{q}_1^0$ (top) and between the internal part of the RC index ($RC^i$) and the induced coefficient $i_1^0$ (bottom), for the period March-July 2009, during the geomagnetically quiet times. The $3\sigma$ interval derived from the model post-variances is shown as a shaded area in plum.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8. : Annual correlation coefficient between the hourly model coefficients and the Dst and RC index during geomagnetically quiet times. The red (resp. blue) curves present the correlation between the external coefficient $\tilde{q}_1^0$ and the external part of the RC index ($RC^e$) (resp. the external part of the Dst, i.e. Est). The pink (resp. cyan) curves present the correlation between the induced coefficient $i_1^0$ and the internal part of the RC index ($RC^i$) (resp. induced internal part of the Dst, i.e. Ist). Correlations decrease from 2021 to 2024 when the number of available observatories decreases.</figcaption></figure>
Southward oriented magnetic field. The external Gauss coefficients of degree and order 1 otherwise point towards the Sun ($q_1^1$ negative) and dawn ($q_1^{-1}$ positive). The correlation between the By component of the Interplanetary Magnetic Field (IMF) and the magnetospheric field coefficients (Lesur et al., 2005) is indicative of the constraints applied on the magnetosphere by the IMF horizontal components, the so-called "penetrating magnetic field" (e.g. Tsyganenko, 2002). The correlation between the IMF By component and the magnetospheric coefficient $q_1^{-1}$ time series is found to be approximately -0.16. The correlation between the IMF Bx component and the magnetospheric coefficients is not clear in our results.

The values derived from our model are slightly different from Maus & Lühr (2005). This is probably because they used satellite data instead of ground observatory data, as in our model. The difference could originate from the fact that satellite data are not sensitive to local crustal biases, that they do not measure the ionospheric field at the same altitudes as observatories, and that they only cover very few local time sectors at the timescales of interest in our study.

# 5 Discussion

## 5.1 Fourier Temporal spectra

We analyze the variations of the output model during quiet conditions using the discrete Fourier transform, as shown in Figure 9. The Fourier temporal spectra of time series from 1st January 1996 to 30th October 2024 covers the range from sub-diurnal to solar cycle periods and has a Nyquist frequency of $1.4 \cdot 10^{-4}$ Hz. The gaps of the model series due to active periods ($|Dst| > 30$ nT) were linearly interpolated.

The top of the Figure 9a (resp. bottom) shows the degree by degree temporal spectra of the SH spatial power of the primary $R^e(\ell)$ and induced $R^i(\ell)$ magnetospheric model at the Earth's surface described by Lowes (1974) and defined as:

$$R^e(\ell) = \ell \sum_{m=-\ell}^{\ell} (q_{\ell}^m)^2, \qquad (19)$$

$$R^i(\ell) = (\ell + 1) \sum_{m=-\ell}^{\ell} (i_{\ell}^m)^2, \qquad (20)$$

The spatial spectrum is independent of the Earth's coordinate system. Figure 9b shows the temporal spectra of the estimated ionospheric perturbations at the Honolulu observatory for northward, eastward
<table><caption>Table 5. Gauss coefficient averages from 1st January 1996 to 30th October 2024 for SH degrees and orders up to 6. First column is the SH degree, the second column is the SH order. The primary magnetospheric field is expressed in the SM (Solar Magnetic) system of coordinates and the induced field in the ECEF (Earth-centered Earth-fixed) system of coordinates. The units are nT. Absolute values smaller than $10^{-4}$ nT have been set 0.</caption><thead><tr><th rowspan="2">ℓ</th><th rowspan="2">m</th><th colspan="2">Magnetosphere (SM)</th><th colspan="2">Induced field (ECEF)</th></tr><tr><th>q<sub>ℓ</sub><sup>m</sup></th><th>q<sub>ℓ</sub><sup>-m</sup></th><th>i<sub>ℓ</sub><sup>m</sup></th><th>i<sub>ℓ</sub><sup>-m</sup></th></tr></thead><tbody><tr><td>1</td><td>0</td><td>6.0214±9.4300</td><td></td><td>0.1453±2.6479</td><td></td></tr><tr><td>1</td><td>1</td><td>-1.542±7.1725</td><td>2.6986±3.8327</td><td>-0.0497±1.5580</td><td>-0.2856±2.1517</td></tr><tr><td>2</td><td>0</td><td>2.1618±3.2520</td><td></td><td>-0.154±1.2854</td><td></td></tr><tr><td>2</td><td>1</td><td>-0.7575±2.3658</td><td>0.953±1.9336</td><td>0.0105±1.5610</td><td>0.1285±1.5731</td></tr><tr><td>2</td><td>2</td><td>1.3388±1.8102</td><td>1.0293±3.7831</td><td>-0.1759±0.9952</td><td>-0.1735±1.0317</td></tr><tr><td>3</td><td>0</td><td>0.5583±1.1274</td><td></td><td>-0.1221±0.2094</td><td></td></tr><tr><td>3</td><td>1</td><td>0.5963±1.0952</td><td>-0.1998±0.5862</td><td>-0.0116±0.2145</td><td>0.04±0.2707</td></tr><tr><td>3</td><td>2</td><td>0.4756±0.8513</td><td>0.2357±0.7506</td><td>0.0271±0.2893</td><td>0.0182±0.2729</td></tr><tr><td>3</td><td>3</td><td>0.6494±1.3357</td><td>-0.1673±0.6863</td><td>-0.0408±0.1830</td><td>-0.0285±0.1894</td></tr><tr><td>4</td><td>0</td><td>-0.0733±0.2106</td><td></td><td>-0.007±0.0107</td><td></td></tr><tr><td>4</td><td>1</td><td>0.32±0.3261</td><td>-0.1389±0.1536</td><td>0.0017±0.0146</td><td>-0.0045±0.0151</td></tr><tr><td>4</td><td>2</td><td>0.0425±0.3509</td><td>0.0077±0.1448</td><td>0.0049±0.0108</td><td>-0.0004±0.0098</td></tr><tr><td>4</td><td>3</td><td>-0.013±0.2310</td><td>-0.0375±0.1260</td><td>0.0003±0.0103</td><td>0.0012±0.0101</td></tr><tr><td>4</td><td>4</td><td>0.0212±0.1471</td><td>-0.114±0.1388</td><td>-0.0006±0.0092</td><td>0.0008±0.0096</td></tr><tr><td>5</td><td>0</td><td>-0.0014±0.0076</td><td></td><td>0.0002±0.0004</td><td></td></tr><tr><td>5</td><td>1</td><td>-0.003±0.0165</td><td>-0.0073±0.0102</td><td>-0.0001±0.0005</td><td>-0.0002±0.0004</td></tr><tr><td>5</td><td>2</td><td>0.0141±0.0209</td><td>0.0064±0.0145</td><td>-0.0001±0.0005</td><td>-0.0±0.0004</td></tr><tr><td>5</td><td>3</td><td>-0.0037±0.0126</td><td>0.0056±0.0202</td><td>0.0001±0.0004</td><td>0.±0.0004</td></tr><tr><td>5</td><td>4</td><td>0.008±0.0133</td><td>-0.0063±0.0137</td><td>-0.0002±0.0004</td><td>0.0001±0.0005</td></tr><tr><td>5</td><td>5</td><td>0.0086±0.0083</td><td>0.0022±0.0107</td><td>0.±0.0004</td><td>0.0001±0.0003</td></tr><tr><td>6</td><td>0</td><td>0.0001±0.0003</td><td></td><td>0.±0.</td><td></td></tr><tr><td>6</td><td>1</td><td>-0.0±0.0006</td><td>-0.0002±0.0003</td><td>-0.±0.</td><td>0.±0.</td></tr><tr><td>6</td><td>2</td><td>0.0004±0.0009</td><td>-0.0003±0.0007</td><td>-0.±0.</td><td>0.±0.</td></tr><tr><td>6</td><td>3</td><td>0.0001±0.0006</td><td>0.0007±0.0008</td><td>0.±0.</td><td>0.±0.</td></tr><tr><td>6</td><td>4</td><td>-0.0004±0.0009</td><td>-0.0005±0.0006</td><td>-0.±0.</td><td>-0.±0.</td></tr><tr><td>6</td><td>5</td><td>0.0005±0.0007</td><td>0.0003±0.0006</td><td>0.±0.</td><td>0.±0.</td></tr><tr><td>6</td><td>6</td><td>-0.0001±0.0004</td><td>0.0002±0.0004</td><td>0.±0.</td><td>0.±0.</td></tr></tbody></table>

and downward components, respectively.

The magnetospheric magnetic field in the near-Earth magnetic environment attributed to the ring current is the most energetic contribution to the magnetic field of external origin. The amplitude of the spectrum for the first SH degree is therefore the largest compared to the other SH degrees for both the magnetospheric field and the internally induced field. The main frequency peaks of all Fourier temporal spectra within the analyzed series are listed in Table 6. Other less prominent peaks are present in the Fourier spectra, showing in particular a dependency of the model to lunar tides, at periods close to $S_1$,
$S_2$, $S_3$ and $S_4$.

<table><caption>Table 6. : Geomagnetic tidal constituents defined in the Figure 9 and discussed in the text.</caption><thead><tr><th>Species</th><th>Darwin Symbol</th><th>Period</th><th>Period in Love & Rigler (2014)</th></tr></thead><tbody><tr><td>Sunspot solar cycle</td><td>R<sub>sun</sub></td><td>~9.6105 yr</td><td>~10.4400 yr</td></tr><tr><td>Principal solar annual</td><td>S<sub>a</sub></td><td>363.1264 d</td><td>365.2422 d</td></tr><tr><td>Principal solar semi-annual</td><td>S<sub>sa</sub></td><td>181.5632 d</td><td>182.6211 d</td></tr><tr><td>Lunar fortnightly</td><td>M<sub>f</sub></td><td>13.6585 d</td><td>13.6608 d</td></tr><tr><td>Principal solar diurnal</td><td>S<sub>1</sub></td><td>23.9992 hr</td><td>24.0000 hr</td></tr><tr><td>Principal solar semidiurnal</td><td>S<sub>2</sub></td><td>12.0002 hr</td><td>12.0000 hr</td></tr><tr><td>Principal solar terdiurnal</td><td>S<sub>3</sub></td><td>8.0000 hr</td><td>8.0000 hr</td></tr><tr><td>Principal solar quarter-diurnal</td><td>S<sub>4</sub></td><td>6.0000 hr</td><td>6.0000 hr</td></tr></tbody></table>

The 11-year solar cycle, denoted $R_{sun}$ and characterized by maxima and minima of relative sunspot numbers (Vaquero & Vázquez, 2009), is identified in the Fourier spectra. However, it lies approximately at the period of 9.6105 yr (~ 10.44 yr, found by (Love & Rigler, 2014) with the Honolulu observatory data with a 0.1 day period resolution). This shift is explained by the period sampling for which the two nearby sampled periods are 14.49 yr and 9.61 yr, so that the solar cycle variation peak seems to concentrate at the closest sampled period of 9.61 yr.

In the Fourier spectra, the annual ($S_a$, 1 cycle per year) and daily ($S_1$, 1 cycle per day) variation peaks are much less pronounced for the magnetospheric and internally induced spatial spectra than for those of the estimated ionospheric perturbations at Honolulu in the ECEF coordinate. The semi-annual ($S_{sa}$, 2 cycles per year) and seasonal (3 or 4 cycles per year) variation peaks of the magnetospheric and internally induced spatial energy are almost similar, with the exception that the seasonal variation maximum of the induced part is not very significant. The semi-annual variation in geomagnetic activity has long been recognized and is considered to be due to the Russell-McPherron (R-M) effect (Russell & McPherron, 1973; Orlando et al., 1993) and to the equinoctial hypothesis effect (E. Cliver et al., 2000; E. W. Cliver et al., 2001; E. Cliver et al., 2002). The R-M effect is the result of variations in the Earth's orbital and rotational patterns within the solar coordinate system (Lockwood et al., 2016, see Figure 1). It explains much more pronounced geomagnetic disturbances near the equinoxes, attributable to the increased geoeffectiveness of the southward interplanetary magnetic fields (IMF) during these times. The R-M effect is clear in the Dst index during periods of both minor and moderate geomagnetic activities. Furthermore, its influence weakens in case of major geomagnetic storms (Lockwood, Owens, et al., 2020).
The equinoctial hypothesis is founded upon the variation of the angle between the Earth-Sun line and the Earth's dipole axis. The effect of the equinoctial hypothesis, associated with the dynamic pressure of the solar wind, appears to have a greater influence on the semiannual variation of the Dst index and of the *am* index during substorm and storm times (E. W. Cliver et al., 2001; Lockwood, McWilliams, et al., 2020). The semi-annual variation of the estimated magnetospheric power and internally induced power in this study is likely to be a consequence of the R-M effect because storm times are not considered by the data selection we made to build the model. The R-M effect also links to annual, seasonal, and diurnal variations (Zhao & Zong, 2012; Lockwood, Owens, et al., 2020). It should be noticed that the amplitude of the annual peak in the Fourier spectrum, when compared with the semi-annual and diurnal variation, is much larger. As is discussed in more details in the next subsection, this could indicate contributions from tail currents to the magnetospheric field model (Maus & Lühr, 2005; Lühr et al., 2017).

In the Fourier spectra of the ionospheric perturbations (Figure 9b), the peaks of semi-annual and seasonal variation gradually weaken up to the point where they vanish compared to their annual variations. In previous studies (e.g. Yamazaki & Maute, 2017), the daily variations of the ionospheric perturbations are considered to be due to the ionospheric dynamo currents, the so-called Solar quiet variations (Sq). It is also suggested that the ionospheric perturbations annual period, and its harmonics, could be impacted by the Sq current system. The study of Liu et al. (2024) shows that the observed ionospheric annual variation is also affected by the observatory positions and the geomagnetic tilt angle. Furthermore, the semi-annual variation is strongly influenced by solar radiations. It therefore seems reasonable that, in the Fourier spectra, the amplitude of the daily variation of ionospheric disturbances is the most significant, while the amplitudes of annual, semi-annual and seasonal variations differ.

Spectral peaks at quasi-monthly periods, including their harmonic oscillations, are observed in the magnetospheric and internally induced spectra (Figure 9a) but less visible or even absent in the ionospheric spectra (Figure 9b). These signals are associated with the approximately 27-day harmonic in terrestrial magnetic fields, as first observed by Bartels (1932). They originate from mechanisms related to solar rotation, which has a synodic Carrington period of 27.2753 days as originally estimated by Carrington (1863) and of ~27.7 days as given in Lockwood & Owens (2024). The double peaks at 25.23 d and 29.33 d in Figure 9a (not evident in Figure 9b) are mainly attributable to variations in the rotational velocity at the solar surface. These peaks are consistent with the analysis of the interplanetary magnetic field
(IMF) in cycle 24, which exhibits pronounced spectral peaks at 26 and 30 d (Chowdhury et al., 2015). Double peaks were also observed in Min & Grayver (2025, see their Figure 17), where the authors modeled external sources and their induced geomagnetic fields (periods at 25.5 d and 29.7 d). It is also observed that additional spectral peaks emerge at harmonics of the Carrington period. A fortnightly oscillation at 14.77 d, more visible in the magnetospheric spectra, is likely a 2nd harmonic of the ~27 d Carrington period. This is also observed at 14.4 d in the work of Min & Grayver (2025) and at 14.7653 d in Love & Rigler (2014). A 3rd harmonic of the Carrington period at 9.01 d is observed in the magnetospheric spectra and internally induced spectra, but not in ionospheric spectra. The variation due to the lunar fortnightly tide, denoted by $M_f$, can only be observed in ionospheric spectra.

## 5.2 Annual, semi-annual and diurnal variation

We now extract the magnetospheric contributions corresponding to annual, semi-annual and diurnal variations from the Fourier spectra, using an inverse Fourier transform. This enables the prediction of these specific contributions in the spatial domain and the analysis of their features in the time domain.

Figure 10 shows the distribution of the estimated magnetospheric field corresponding to the semi-annual variations ($S_{sa}$=181.5632 d, in Table 6) in the SM coordinate system. The right panel of the figure shows the relative amplitudes and phases of the semi-annual variations for each Gauss coefficient up to SH degree 4. The coefficients of SH degree 5 and 6 are not shown in the right panel, as their values are too small to be clearly visible. Moreover, we discussed above that the modeling method hardly extracted any significant information from the data for these SH degrees. The Gauss coefficients have different wave phases for semi-annual variations. We notice that the axial dipole component $q_1^0$ dominates and has maximum values around the equinoxes. It links to the early study of Banks (1969) that proposed that the change in the ring current intensity affects the large-scale dipole $q_1^0$ component for the semi-annual variations of the magnetosphere. The maps centered at noon in the left and center panels of the figure present the spatial distribution of the spherical northward component of the magnetospheric field at Earth's surface in the SM coordinate. In the left panel, when the $q_1^0$ component is maximum, the map shows a global southward magnetospheric field that is most likely produced by the westward ring current
<figure><img src="image_11.png" /><figcaption>Figure 9. : Fourier spectra of modeled external and induced field contributions (in days). (a) Time spectra of the spatial power of the modeled external Gauss coefficients $q_{\ell}^m$ (top) and modeled internal coefficients $i_{\ell}^m$ (bottom) at Earth's surface, shown by SH degree. (b) Time spectra of estimated ionospheric perturbations at Honolulu for northward (x), eastward (y), and downward (z) components. Main spectral periods are listed in Table 6.</figcaption></figure>
and also two maxima of the field on the day side above and below the equator. In the center panel, when $q_1^0$ is around zero, the components $q_1^{\pm 1}$ and $q_2^0$ dominate the weak remaining signal. The semiannual modulation of the magnetospheric signal, marked by two maxima above and below the equator on the day side and only one maximum on the night side, bears some similarities with the expected signature of the cut-ring current, which, as a part of the ring current, does not go around the Earth as one system but splits into two branches on the day-side magnetosphere (Ganushkina et al., 2018). It might also be the signature of un-modeled ionospheric perturbations.

Figure 11 shows the same as Figure 10, but for diurnal variations. In the right panel, the Gauss coefficients have different wave phases for diurnal variations. The Gauss coefficient quadrupole $q_2^1$ component of the magnetospheric field is the largest followed by $q_1^{\pm 1}$ component. The larger quadrupole term in the SM coordinate system represents a local time dependence of the ring current intensity, as earlier reported by Langel et al. (1996) and also observed in induced field studies (Balasis et al., 2004). Due to the link between R-M effect and magnetosphere variation described above in sub-Section 5.1, we look at the correlation between the quadrupole model component and the IMF Bz component. It is found to be -0.18, which means that this component has been affected to some extent by the R-M effect. In the maps of the extracted northward magnetospheric field, the $q_2^1$ component dominates the signal in the left panel, and in the center panel when the $q_2^1$ component is around zero, the components $q_1^{\pm 1}$ and $q_2^0$ dominate the signal. The patterns of the field above and below the equator at both noon and midnight are consistent with the signals observed by Lühr et al. (2017) that were interpreted as signatures of magneto-tail currents. However, in their study, the patterns are produced by variations of the external SH degree 1 and order 1 coefficient with local time in the GSM coordinate system, whereas, we consider the field produced by all coefficients in the SM coordinate system.

Figure 12 presents the magnetospheric signal associated with annual variations ($S_a$) in the SM spherical coordinate system. According to previous studies, (e.g. Malin & Işikara, 1976; Hibberd, 1985; Wardinski & Mandea, 2006), this signal could be due to a north-south motion of the ring current that affects the $q_2^0$ component. In the right panel, the $q_2^0$ component is large, but the $q_1^1$ component that describes the day-night asymmetry is the largest. It reaches up to 13.06 nT. In the CHAOS-5 model (C. C. Finlay et al., 2015), during quiet geomagnetic activity, annual variations of the magneto-tail signal, estimated from Ørsted and CHAMP satellite data, are characterized by variations of the $q_1^1$ component of about 12 nT (Lühr et al., 2017). It seems therefore that both the annual motion of the ring current and the tail
currents contribute to the observed annual variations of the magnetospheric signals.

We have seen in Figure 10,11,12 that a few Gauss coefficients characterize the behavior of magnetosphere variations. We therefore investigate now how these coefficients average over years and months. In Figure 13, the $q_1^0$ coefficient (in dark blue) of the magnetospheric field model is linked to changes in the ring current intensity. It presents a solar-cycle dependence and a semi-annual dependence with maximum near the equinoxes and minimum at solstice. The $q_2^0$ coefficient (in light blue) does not depend on solar cycle but shows an annual variation that reaches a maximum during the north hemisphere summer and a minimum during the north hemisphere winter. The $q_1^1$ (in green) coefficient does not depend on solar cycle either but shows an annual variation with anti-phase during summer and winter, and the value around the equinoxes is close to zero, just as described in Lühr et al. (2017).

The above analyses demonstrate that the variations of the magnetospheric magnetic field model are associated with the changes in both the ring current and the magnetotail currents, due to the interplanetary magnetic field interacting with the magnetosphere. There is, however, no evidence of significant contamination from ionospheric sources.

## 5.3 Average local time asymmetry

In previous studies (e.g. Maus, 2015; Lühr et al., 2017; Lühr & Zhou, 2020), it was found that the poor description of magnetospheric behaviors, such as asymmetric structure and local time dependence, could significantly affect geomagnetic field models describing the main core field, even if these are generally built using data collected during magnetically quiet times. The local-time asymmetry (dawn-dusk asymmetry) is thought to come from the inhomogeneous distribution of ring current intensity. With in situ measurements, the ring current intensity (Q.-H. Zhang et al., 2011, Figure 4 with $Dst > -30nT$) and the density of the energetic neutral atom in the ring current (C:son Brandt et al., 2002, Figure 1 during storm times), at a distance of 2 to 8 Earth radii, were found to be the largest in the post-midnight and early morning sectors and to be the smallest in the afternoon sector. However, other studies based on low-altitude satellite and ground observatory data (Lühr et al., 2017, Figure 5 and 6) (Lühr & Zhou, 2020, Figure 4) (Fillion et al., 2023, Figure 8 and 9) find that the dawn-dusk asymmetry is stronger in the dusk than in the dawn sectors near the equatorial plane during storm times at ground level. In these studies, the authors conclude that the ring current may not be the primary cause of
<figure><img src="image_12.png" /></figure>

Figure 10. : Spatial and temporal behaviors of the modeled magnetosphere contributions corresponding to the semi-annual ($S_{sa}$) variations. The left and center panels show the map of the magnetosphere field for northward component at the Earth's surface. The maps are centered at noon in the SM coordinate system. The pink lines indicate dawn-dusk local times - i.e. 06h00 and 18h00. The right panel shows the phases and amplitudes of each modeled Gauss coefficients (up to SH degree 4).

<figure><img src="image_13.png" /></figure>

Figure 11. : Same as Figure 10 but for the $S_1$ variations.

<figure><img src="image_14.png" /></figure>

Figure 12. : Same as Figure 10 but for the $S_a$ variations.
<figure><img src="image_15.png" /><figcaption>Figure 13. The distribution of the annual averages (left panel) and monthly averages (right panel) of $q_1^0$ (in dark blue), of $q_1^1$ (in green) and of $q_2^0$ (in light blue), from 1st January 1996 to 30th October 2024 during geomagnetically quiet time.</figcaption></figure>
dawn-dusk asymmetry. Rather, it is suggested that the partial ring current and field-aligned current may be the underlying factors contributing to the dusk sector being stronger and the dawn sector weaker magnetospheric field intensities. No clear evidence for such magnetospheric asymmetry is found during geomagnetically quiet conditions as it is easily masked by ionospheric perturbations (Fillion et al., 2023). Therefore, it is interesting to discuss how the local-time asymmetry is seen in our model.

The average southward component of the modeled magnetospheric field at ground level and at the magnetic equator was computed, during quiet times from 1st January 1996 to 30th October 2024. The field is expressed in SM magnetic local time (SMLT) and split into contributions from the zonal terms only (Figure 14) and from the non-zonal terms only (Figure 15a). The ring current signal influences the most the zonal terms of the estimated Gauss coefficients. Figure 14 indicates an averaged southward field (positive sign), the colorful lines from dark to light show levels of solar activity from low to high as given by solar flux level. The increase of solar activity enhances the ring current intensity, which in turn increases the southward magnetospheric field.

Figure 15a shows the large-scale asymmetric magnetospheric field, i.e. the field truncated to SH degree 2 with zonal terms set to zero. An average local-time asymmetry is observed at all levels of solar activity. The peak-to-peak amplitude of the asymmetry reaches approximately 4 nT, showing a northward field near the dawn sector and a southward field near the dusk sector. This pattern is consistent with the dawn-dusk asymmetry analyzed at ground level in Fillion et al. (2023, Figure 7). When the solar activity level is high, the local-time asymmetry keeps the same structure as when the solar activity level is low, but the maxima occur at 9 SMLT and 21 SMLT. When solar activity reaches an intermediate level, the local-time maxima shift to around 6 SMLT and 18 SMLT. The amplitude of the asymmetric field is the largest during solar minimum activity, and it is smallest during maximum solar activity. This may be an indication that this asymmetry is produced not only by the ring current, even during quiet times.

For the internally induced field at the magnetic equator (Figure 15b), the dawn-dusk asymmetry is not found during quiet times, the maxima are always found to be at noon and midnight as already described in Fillion et al. (2023, Figure 7). In addition, the amplitudes of the non-zonal terms for the internally induced field increase when the solar activity level increases and slightly shift in local time. We recall that the induced field is the result of all external magnetic field variations, including both the ionosphere and the magnetosphere fields. Here, it is likely that, as explained in Fillion et al. (2023), the internally induced field model for the non-zonal terms describes the field mainly induced by ionospheric fields.
<figure><img src="image_16.png" /><figcaption>Figure 14. : Southward component of modeled external zonal magnetospheric fields (1 Jan 1996–30 Oct 2024) at the magnetic equator in Solar Magnetic (SM) coordinates.</figcaption></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>Figure 15. : Average asymmetric southward component of modeled magnetic fields (1 Jan 1996–30 Oct 2024) at the magnetic equator in Solar Magnetic (SM) coordinates for fields from non-zonal terms: left panel shows averages versus SM local time (SMLT) and solar activity, right panels show amplitude profiles with SMLT for selected solar activity levels.</figcaption></figure>
## 5.4 Equivalent sheet of currents

To gain insight into the spatial configuration and temporal dynamics of the estimated external field, we also present the results using stream-functions of equivalent currents, as defined in, e.g. Lesur et al. (2008), see Eq. (14) and (15); Min & Grayver (2025), see Eq. (23).

Figure 16 shows the average southward component of the estimated magnetospheric field and the equivalent current (in black arrow) at 100 km above the reference sphere – i.e. at a radius of 6471.2 km, for the solar activity at low ($F10.7 \le 120. \text{ sfu}$), moderate ($120. \le F10.7 < 170. \text{ sfu}$) and high ($F10.7 \ge 170. \text{ sfu}$) levels respectively. The black arrows represent the horizontal direction and strength of the equivalent current. We notice that the westward ring current dominates for the three solar activity levels and that the intensity of the equivalent ring current increases as the level of solar activity rises. The equivalent current is more intense in the morning sector for all three cases. In addition, in the morning sector, a northward component (in blue) of the average magnetospheric field around 5 nT is visible at low solar activity level, but when the solar activity level becomes stronger and the ring current becomes stronger, the northward component is masked by the ring current signal.

To understand this, the southward component of the average modeled magnetospheric field with its SH zonal terms subtracted is shown in Figure 17 for the three different levels of solar activity. The northward component (in blue) in Figure 16 in the morning sector is visible in Figure 17 at the three levels of solar activity. The values increase slightly with the increase in the level of solar activity, not as much as the ring current signal. The patterns, shown in Figure 17, are similar to the expected structure of the magnetotail current signal. Figure 18 shows the averaged annual behaviors of the modeled non-zonal northward magnetospheric field (left panel) and the modeled non-zonal eastward magnetospheric field (right panel) at midnight and 6 SM local time respectively. The results are in good agreement with those obtained by Lühr et al. (2017, Figure 12) that describe an annual variation caused by the magneto-tail currents.

We conclude that the magneto-tail currents may contribute to the non-zonal magnetospheric field signals and that the results presented in Figure 16 are explained by the contributions of both ring and magneto-tail currents.
<figure><img src="image_19.png" /></figure>

Figure 16. : The maps of the average southward magnetosphere field and of the equivalent current in black arrows at 100 km altitude in SM coordinate during geomagnetically quiet times from 1st January 1996 to 30th October 2024. From left to right: the map centered at noon at solar activity from low level (F10.7<120.sfu) to high level (F10.7≥170.sfu).

<figure><img src="image_20.png" /></figure>

Figure 17. : The maps of the non-zonal average southward magnetosphere field at 100 km altitude in SM coordinate during geomagnetically quiet times from 1st January 1996 to 30th October 2024. From left to right: the map centered at noon at solar activity from low level (F10.7<120.sfu) to high level (F10.7≥170.sfu).

<figure><img src="image_21.png" /></figure>

Figure 18. : Annual properties of the non-zonal modeled magnetosphere field at Earth's surface. Left: the northward components in SM coordinate at midnight for different latitudes and for different months. Right: the eastward components in SM coordinate at 06 SMLT for different latitudes and different months.
## 6 Conclusion

This study presents a model of the magnetospheric contribution to the geomagnetic field over nearly three decades (1996.0 to 2024.8) that has been built using high-resolution ground-based geomagnetic observatory data during geomagnetically quiet periods. Using a Kalman filter-based modeling framework with a correlation-driven data assimilation technique, we developed a time series of hourly SH models up to degree and order 6, capturing both external magnetospheric and internally induced fields.

The resulting models demonstrate robust performance, with residual standard deviations values lower than about 5 nT for most observatories (see Figure 19), except in high-latitude regions affected by auroral electrojets. Independent validation using data from the SOK observatory confirms the model's accuracy, with residuals consistent with expected noise levels. The estimated external and internal dipole components exhibit good agreement (i.e. correlation exceeding 0.8) with established geomagnetic indices such as RC and Dst, validating the modeling framework. Analysis of the posteriori variances shows a strong dependence on the available number of observatories, it also shows that information can be extracted from observatory data up to SH degree 4. Beyond this scale, the models have a limited resolution but SH degrees larger than 4 were kept to minimize possible truncation effects and aliasing.

Spectral analyses reveal well-defined periodicities, including diurnal, semi-annual, annual, and quasi-monthly variations, as well as solar-cycle effects, consistent with known mechanisms such as the Russell-McPherron effect and Carrington rotation. Moreover, the spatial-temporal structures of the derived models confirm the expected evolution in time and configuration of magnetospheric signals, particularly those associated with the ring currents and magneto-tail currents. They highlight the dependence of the signals on local time, season, and solar activity. The modeled magnetospheric fields also capture subtle local time asymmetries and confirm results from existing models regarding magnetospheric dynamics during quiet conditions. The internally induced fields, however, do not exhibit the same asymmetry, suggesting that the dominant non-zonal induced fields may be due to ionosphere magnetic field variations.

This work provides insight into magnetospheric dynamics during quiet conditions and relies on magnetic observatory data. However, separating internal and external field contributions using ground-based measurements alone remains challenging. Therefore, we acknowledge that the accuracy of this separation currently depends on the crustal and ionospheric a priori statistical models embedded in the Kalman filter.
As a result, some constant leakage from the crustal field, into the primary magnetospheric coefficients for example, and some time-varying leakage associated with ionospheric fields at typical diurnal and seasonal periods, cannot be fully ruled out. Ongoing efforts is currently made to incorporate Swarm satellite observations to better assess the internal/external field separation achieved in this study and to extend the analysis to geomagnetically active periods. These developments are also built on recent and forthcoming advances in satellite magnetic field monitoring, including the MSS-1 mission (K. Zhang, 2022) and the planned NanoMagSat ESA Scout mission (Hulot et al., 2020).

## Appendix A. Data misfits of the observatories at low- and mid-latitudes

In this appendix, Figure 19 presents the misfits between observed and modeled vector field components at each INTERMAGNET observatory across low to mid-latitudes (below Quasi-Dipole latitude 60°). The mean residuals (star symbols) lie close to zero, indicating minimal bias between model and data, and the standard deviations (horizontal lines) show that residuals generally fall within ±5 nT.

## Data availability statement

The input hourly ground observatory data are from INTERMAGNET and WDC and prepared by the British Geological Survey, accessible via ftp://ftp.nerc-murchison.ac.uk/geomag/Swarm/AUX OBS.

The SOK observatory data are available from magnetic databank BCMT via http://dx.doi.org/10.18715/BCMT.MAG.VAR. The MCM core magnetic field model subtracted from the data is:

SW_OPER_MCO_SHA_2Y_19990101T000000_20240731T000000_1101.SHC

archived at https://swarm-diss.eo.esa.int/#swarm%2FLevel2longterm%2FMCO. The Kalmag core field model can be obtained at https://ionocovar.agnld.uni-potsdam.de/Kalmag/Kalmag/Model/.

The RC indices are provided in the CHAOS-8.2 model package available at https://www.spacecenter.dk/files/magnetic-models/CHAOS-8/latest.html. Dst indices before the year of 1998 are available from https://isgi.unistra.fr/data_download.php and after 1998 are available from https://earth.esa.int/eogateway/missions/swarm/data. F10.7 indices are downloaded from https://omniweb.gsfc.nasa.gov/form/dx1.html or https://spdf.gsfc.nasa.gov/pub/data/omni/low_res_omni/. The set ionosphere perturbations are derived from runs of the TIEGCM simulation and available at Zenodo via Maute (2020). The time series of estimated parameters are available at https://github.com/yalei-SHI/Hourly_model_Magnetosphere_Induced.git, or through IPGP Research Collection
Data misfits for low- and mid-latitude observatories

<figure><img src="image_22.png" /></figure>
Data misfits for low- and mid-latitude observatories

<figure><img src="image_23.png" /></figure>

Figure 19. : Means residuals (star symbols) and standard deviations (horizontal bars) for all INTER-MAGNET observatories at low- to mid-latitudes (below Quasi-Dipole latitude 60°), from 1st Jan 1996 to 30th Oct 2024 during geomagnetically quiet time, for the northward (left), eastward (middle), and downward (right) components, respectively.
(research-collection.ipgp.fr) at the following DOI: http://dx.doi.org/10.18715/IPGP.2025.mfmhsm39
(Note that this DOI will be activated upon acceptance of the manuscript).

## Abbreviations

RC: Ring Current; LS: Least Squares; Dst: Disturbance storm time; SH: spherical harmonic; Obs: observatory; TIEGCM: Thermosphere-Ionosphere-Electrodynamics General Circulation Model; IMF: Interplanetary Magnetic Field; R-M effect: Russell-McPherron effect; SM: Solar Magnetic (reference frame); ECEF: Earth-centered Earth-fixed (reference frame); GSM: Geocentric Solar Magnetic (reference frame). *sfu*: Solar flux unit ($10^{-22} Wm^{-2}Hz^{-1}$)

## Competing interests

The authors declare that they have no competing interests.

## Funding

This study was partly funded by the Centre National des Etudes Spatiales (CNES). PhD funding of YS was jointly provided by CNES and Ecole Doctorale ED 560 STEP'UP Université Paris Cité.

## Authors' contributions

VL and ET proposed this research project and the modeling method. YS developed the code, derived and analyzed the model with contributions from VL and ET. YS wrote the manuscript with contributions from VL and ET. All authors participated in the discussion and commented on the manuscript. All authors read and approved the final manuscript.

## Acknowledgments

We would like to thank the Centre National des Etudes Spatiales (CNES) who partly funded this study. The results presented in this paper rely on data collected at magnetic observatories. We thank the national institutes that support them and INTERMAGNET for promoting high standards of magnetic observatory practice (www.intermagnet.org). We also thank the British Geological Survey
(BGS) for making available a quasi-real time compilation of INTERMAGNET data in the framework of the European Space Agency "Swarm DISC (Swarm Data, Innovation, and Science Cluster)" international consortium. We thank A. Maute for providing the ionosphere model TIEGCM.

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