# Next-Generation Data Assimilation Methods for Polar Ionospheric Electrodynamics

Karl M Laundal, Aurélie Marchaudon, Astrid Maute, Spencer M Hatch, Florine Enengl, Tomoko Matsuo, Margot Decotte, Michael Madelaire, Viacheslav G Merkin, Anthony Sciola, et al.

► To cite this version:

Karl M Laundal, Aurélie Marchaudon, Astrid Maute, Spencer M Hatch, Florine Enengl, et al. Next-Generation Data Assimilation Methods for Polar Ionospheric Electrodynamics. Surveys in Geophysics, inPress, 10.1007/s10712-025-09918-3. hal-05441681

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# Next-Generation Data Assimilation Methods for Polar Ionospheric Electrodynamics

Karl M. Laundal¹,² ID · Aurélie Marchaudon³ · Astrid Maute⁴ · Spencer M. Hatch² ·

Florine Enengl⁵ · Tomoko Matsuo⁶ · Margot Decotte² · Michael Madelaire¹ ·

Viacheslav G. Merkin⁷ · Anthony Sciola⁷ · Veronika Haberle⁸ · Andreas S. Skeidsvoll²

Received: 7 April 2025 / Accepted: 18 November 2025

© The Author(s) 2025

## Abstract

Accurately specifying polar ionospheric electrodynamics is essential for understanding energy and momentum exchange between space and the upper atmosphere and for improving simulations of the ionosphere and the thermosphere. Statistical models are commonly used to provide input for global circulation models (GCMs). However, maps derived from simultaneous multi-instrument observations better represent the actual state of the system. Such maps integrate measurements from ground-based magnetometers and radars, in situ plasma and magnetic field sensors at low-Earth orbit, and optical and particle observations of auroral precipitation. However, ionospheric data assimilation remains in its early stages. Current methods rely on restrictive assumptions to simplify equations and stabilize inverse problems, but these constraints limit applicability beyond polar regions, hinder the inclusion of time-dependent processes, and prevent independent estimation of ionospheric conductance. This review examines the physical foundations of ionospheric data assimilation, evaluates the limitations of existing approaches, and explores pathways toward more accurate and flexible techniques. Specifically, we discuss approaches to: (1) use a common dataset to estimate conductance and fields in a single inversion; (2) incorporate neutral winds instead of assuming they are zero; (3) account for a realistic main magnetic field geometry instead of assuming radial field lines; (4) eliminate a sharp boundary between polar and low-latitude regions; (5) use F-region density measurements to capture the history of ionospheric conductance and plasma transport; (6) account for the magnetic field of ground-induced currents in a more realistic way; (7) include ionospheric induction effects to stabilize time-dependent inversions; and (8) couple ionospheric electrodynamics with global magnetosphere simulations to model the physics of time variations.

**Keywords** Ionosphere · Ionospheric electrodynamics · Ionospheric data assimilation · Ionospheric measurements
# 1 Introduction

The interaction between the atmosphere and space is arguably most direct in a narrow altitude range of the lower ionosphere, between about 100 and 150 km. This is where momentum and energy exchange between ions and neutrals peaks, as a result of collisional interactions. Particularly in the polar regions and auroral zones, this atmospheric layer serves as a crucial energy sink in geospace, experiencing significant heating (Thayer and Semeter 2004; Palmroth et al. 2021), which in turn leads to pressure gradients that drive neutral dynamics globally (Fuller-Rowell et al. 1994). Moreover, the lower ionosphere is where the atmosphere directly imparts momentum to the space plasma, which is subsequently transferred to the magnetosphere via Alfvén waves, an effect that dominates the plasma dynamics in the inner magnetosphere.

The ionosphere is a region with a relative abundance of observations, from which magnetospheric dynamics is routinely inferred (Milan et al. 2017). Accurate mapping of the dynamics of the lower ionosphere, enabled by the integration of diverse data sources, is essential for understanding the coupling processes between the atmosphere and space plasma. The correct specification of magnetospheric driving is critical to drive models of the upper atmosphere and ionosphere (Qian et al. 2014; Ridley et al. 2006). The input for such models is two-dimensional maps of the electric field implied by the magnetospheric plasma motion, or two-dimensional maps of the field-aligned current (FAC)(Marsal et al. 2012; Maute et al. 2021), along with two-dimensional maps of auroral particle precipitation from which ionospheric conductivities are calculated. Such maps can be produced from a variety of datasets: Maps of the ionospheric convection electric field can be generated directly from ground radar (Chisham et al. 2007) or satellite measurements of plasma flow (Weimer 2005). FAC maps can be generated based on magnetometer data from satellites and Ampère's law (Anderson et al. 2000).

Since magnetic field disturbances and plasma convection are closely related, it is possible to use physics models to combine different types of measurements to yield better accuracy. This combination is the underlying idea of existing data assimilation methods in ionospheric physics (Richmond and Kamide 1988; Lu 2017; Laundal et al. 2022; Matsuo 2020). Existing data assimilation methods fuse magnetic field measurements from space and ground with measurements of the plasma flow, using a range of assumptions that we discuss in more detail below. The output is typically two-dimensional maps of the electric field, plasma velocity, field-aligned and horizontal currents, magnetic field disturbances, and derived quantities such as Joule heating and Poynting flux.

This review examines the physical foundations of ionospheric data assimilation, evaluates the limitations of existing approaches, and explores pathways toward more accurate and flexible techniques. In Sect. 2, we briefly review existing techniques for ionospheric data assimilation. Since they are described in detail elsewhere, we focus here on the underlying principles, the physical assumptions, prior information, and data sources. Section 2 also contains brief reviews of common datasets and empirical models that are useful in ionospheric data assimilation. The subsequent sections form the core of this review by outlining various avenues to improve existing techniques:

Section 3 discusses the critical role of ionospheric conductance. In existing assimilation techniques, the conductance is estimated in a pre-processing step, which allows the magnetic field and plasma velocity to be related through linear equations. Such conductance estimates are typically based on optical measurements of particle precipitation (aurora), empirical models, or a combination of both. In Sect. 3.1, we briefly review the challenges associated with
these approaches. In Sect. 3.2, we discuss recent efforts to use simultaneous measurements of magnetic field disturbances and plasma velocity to co-estimate conductance with other electrodynamical quantities, since these measurements potentially contain additional information about the conductance.

Section 4 discusses assumptions in existing data assimilation techniques that prevent their application beyond very high latitudes. For example, it is common to assume that the neutral wind, whose influence increases toward low latitudes, is zero in the reference frame rotating with the Earth. The implications of this approximation, along with strategies for moving beyond it, are discussed in Sect. 4.1. Furthermore, it is common to assume that the main magnetic field is radial, an assumption that is only reasonable at high latitudes. This assumption, together with approaches for better representing the main magnetic field geometry, is discussed in Sect. 4.2. In addition, the need to impose boundary conditions at low latitudes to constrain the solution complicates the extension of data assimilation techniques to these regions, as further discussed in Sect. 4.3.

Existing ionospheric data assimilation techniques primarily fuse disparate data types to produce maps that satisfy the ionospheric Ohm's law, which represents a steady-state solution to the ion and electron momentum equations. Several important advances in ionospheric data assimilation can be made by incorporating time-dependent processes, which is the topic addressed in Sect. 5. First, potential approaches to include measurements of F-region plasma density are discussed in Sect. 5.1. Because of the long lifetime of F-region plasma density, incorporating these measurements requires accounting for the time history of ionospheric convection (Blelly et al. 2005; Oksavik et al. 2010), plasma production, and decay. A second challenge, discussed in Sect. 5.2, is that existing data assimilation techniques rely on steady-state equations to relate the different quantities. Specifically, Faraday's law is ignored, which can be important in some cases (Madelaire et al. 2024). A third time-dependent aspect in ionospheric data assimilation is ground induction. The magnetic field observed on the ground is related to both ionospheric currents and to currents induced within the Earth. The induced part of the magnetic field has a complex dependence on the time history of the ionospheric magnetic field disturbances and on the three-dimensional conductivity of the Earth (Grayver et al. 2021). To use ground magnetometer data to specify the state of the ionosphere, one must correctly isolate the ionospheric contribution. In Sect. 5.3 we discuss how this contribution has been approximated in the past and offer suggestions for how it can be calculated more accurately in the future. Finally, in Sect. 5.4, we discuss the potential for integrating ionospheric observations into magnetosphere simulations as part of a two-way coupling, where a more realistic magnetosphere feeds back to improve ionospheric data assimilation. Polar ionospheric dynamics is largely affected by processes occurring far away in the magnetosphere. The strong coupling between these two spatially and dynamically distinct regions introduces challenges in ionospheric data assimilation that differ from those in atmospheric data assimilation, where observations and controlling processes tend to overlap more directly.

Finally, in Sect. 6, we discuss other promising directions for further development that go beyond increased physical realism, both in terms of data assimilation algorithms and in how they are used.

## 2 Interrelations of Quantities in Polar Ionospheric Electrodynamics

In this section we outline the physical basis for ionospheric data assimilation and discuss the assumptions used in existing techniques.
The ionosphere is a partially ionized plasma, with charged particles embedded within the neutral gas of the Earth's atmosphere, and its dynamical behavior is primarily governed by electromagnetic interactions and momentum exchange with the neutral fluid through collisions and chemical interactions among ions and neutrals. The relevant physical quantities are the electric field **E**, current density **j**, magnetic field perturbation **ΔB**, plasma velocity **v**, neutral wind **u**, and parameters for the collisional interaction between plasma and neutrals, known as conductivities. Although measurements of each quantity provide insight into the electrodynamic state of the ionosphere, none offers a complete picture on its own. To obtain a more comprehensive view, these measurements can be combined using a model that relates the physical quantities.

The model most commonly used to relate these quantities is the ionospheric Ohm's law, which relates the current density to the electric field in the reference frame of the neutral gas,

$$ \mathbf{j} = \sigma_p(\mathbf{E}_{\perp} + \mathbf{u} \times \mathbf{B}) - \sigma_H(\mathbf{E} + \mathbf{u} \times \mathbf{B}) \times \frac{\mathbf{B}}{B} + \sigma_{\parallel} \frac{\mathbf{E} \cdot \mathbf{B}}{B}. \quad (1) $$

Here $\sigma_p$ and $\sigma_H$ denote the Pedersen and Hall conductivities, respectively, which primarily depend on the plasma density and the ion-neutral collision frequency for momentum transfer. The subscript $\perp$ in the first term indicates the component perpendicular to the magnetic field **B**. The last term, which represents the current parallel to **B**, is usually omitted because the parallel electric field is very low throughout the ionosphere due to the high field-aligned electron mobility (Brekke 2013). Instead, the parallel component of the electric current is determined by taking the divergence of the perpendicular current. The term “law” is somewhat misleading for this equation, since it is valid only under a specific set of restrictive assumptions. In particular, it is assumed that the Lorentz forces on ions and electrons are exactly balanced by their friction with the neutrals, and that all other forces can be neglected. Moreover, for the ionospheric Ohm's law to be useful in data assimilation, additional assumptions must be made. First, the magnetic field **B** is usually represented using a model of the main magnetic field, such as the International Geomagnetic Reference Field (IGRF) (Alken et al. 2021), which means that the effects of magnetic field perturbations are neglected when specifying **B** in Equation (1). Second, because the neutral wind is notoriously difficult to measure and is usually small compared to the plasma velocity in the polar regions, it is often set to zero in a reference frame rotating with the Earth. Third, the electric field is often assumed to be a potential field, $\mathbf{E} = -\nabla\Phi$, which implies that induction is neglected. Furthermore, since few measurements are available at altitudes where $\sigma_p$ and $\sigma_H$ are significant (approximately 100–150 km), we rely on remote measurements, which effectively limits our description of the ionosphere to two dimensions rather than a fully three-dimensional representation. Hence, a height-integrated version of the ionospheric Ohm's law is used,

$$ \mathbf{J} = -\Sigma_p \nabla\Phi + \Sigma_H \left( \nabla\Phi \times \frac{\mathbf{B}}{B} \right), \quad (2) $$

where **J** is a horizontal surface current (in A/m) that represents the height-integrated horizontal component of the volume current density **j** (in A/m²). The quantities $\Sigma_p$ and $\Sigma_H$ are the height-integrated conductivities, known as conductances. Since $\Phi$ is the electrostatic potential, this equation relates the current to the electric field. For the equation to be valid, the magnetic field must be radial, which is a somewhat reasonable assumption at high
latitudes but a poor assumption at mid and low latitudes where a more complex formula-
tion should be used (Richmond 1995; Brekke 2013).

Plasma velocity is most often measured in the F-region, where collisions are less
important and **j** × **B** can be considered small. In this case, plasma velocity is related to
**E** by the generalized Ohm's law for ideal MHD,

$$
\mathbf{E} = -\mathbf{v} \times \mathbf{B}. \tag{3}
$$

Equations (2) and (3) can be used together with Ampère's law, which relates **ΔB** and **J**, to provide the necessary relationships among **E**, **J**, **ΔB**, and **v**.

2.1 Applying the Theory to Relate E, J, ΔB, and v

Here, we discuss in more detail how the equations above are used in practice to relate **E**, **J**, **ΔB**, and **v**. We focus specifically on two published techniques that express ionospheric electrodynamics in terms of the electric field, which is estimated via an inversion that can incorporate both F-region plasma convection and magnetic fields above and below the ionosphere.

The most well-known and established technique for ionospheric data assimilation is the Assimilative Mapping of Ionospheric Electrodynamics (AMIE) technique, developed by Richmond and Kamide (1988); Richmond (1992) (see review by Lu (2017)). It has been extensively used for over three decades and was recently refined (Matsuo 2020) as part of the Assimilative Mapping Geospace Observations (AMGeO) collaborative open-source data science platform (AMGeO Collaboration 2019). More recently, an open-source alternative, the Local Mapping of Polar Ionospheric Electrodynamics (Lompe) technique, was developed by Laundal et al. (2022). The specifics of these methods are outlined below; however, since both are grounded in the same physical assumptions, we first briefly review these common foundations.

Both AMIE and Lompe assume that height-integrated ionospheric Ohm's law (Equation 2) is valid. Plasma velocity measurements in the F-region, where Equation (3) is valid, are incorporated by obtaining the corresponding electric field **E** from Equation (3) and mapping it down to the E-region, where Equation (2) applies, while assuming that the neutral wind is zero. The assumption of zero neutral wind is discussed further in Sect. 4.1. Both techniques also assume that the conductance is known or is fitted in a pre-processing step (see Sect. 3 for more details). With known conductance, the current and electric field are related by linear equations, which simplifies the calculations; however, connecting the magnetic field to the two-dimensional (height-integrated) horizontal current **J** of Equation 2 remains challenging. The primary challenge arises because the magnetic field is fundamentally a global (Biot-Savart) integral over all currents, which suggests that global numerical integration is required at each measurement point. Both AMIE and Lompe avoid this issue by using specific basis functions to represent the current, enabling analytical relationships with the magnetic field. Another challenge is that some parts of the three-dimensional current system are shielded when observed from the ground, whereas satellites may be too high to detect other parts of the current system. Therefore, the regions above and below the lower ionosphere must be treated separately, which directly impacts how magnetic field measurements are used to estimate the electric field.
### 2.1.1 Relating Magnetic Field Measurements to E

The AMIE and Lompe techniques both use the same principles to incorporate magnetic field measurements in estimating the electric field. A key challenge is that the three-dimensional current, which is related to the electric field through the ionospheric Ohm's law, exhibits very different magnetic signatures on the ground compared to in space. Figure 1 illustrates this: The left panel shows a three-dimensional current system, while the right panel presents line plots of the corresponding magnetic field components along the dawn-dusk meridian (as indicated by the thick line in the polar plot) at various radii. For this particular current configuration, the eastward magnetic field component is weak on the ground but strong in space; the northward component reverses sign above and below the ionosphere; and the radial component remains in the same direction but decreases more rapidly with altitude than the horizontal components. These differences in variation above and below the ionosphere must be accounted for to relate the magnetic field to *E* via the current and ionospheric Ohm's law.

The magnetic field on the ground, where there is no electric current, is curl-free and can thus be expressed in terms of a magnetic potential. It is always possible to mathematically construct a sheet current on a spherical surface that produces the observed magnetic field. This hypothetical sheet current, located in the ionosphere, is known as the equivalent current and has been extensively studied for more than a century (Kamide 1982; Untiedt and Baumjohann 1993; Birkeland 1908). The challenge lies in determining how the equivalent current relates to **J** as given by Equation (2). This relation is greatly simplified by assuming that the effects of remote large-scale currents (such as the ring current) are negligible and that the main magnetic field is radial. Under these assumptions, the equivalent current can be interpreted as the divergence-free component of **J**. Helmholtz's theorem implies that **J** can be uniquely decomposed into divergence-free and curl-free components. The equivalent current is divergence-free, so taking the curl of Equation (2) gives an equation in which the components of **J** that are unobservable from the ground are eliminated. This

<figure><img src="image_2.png" /><figcaption>**Fig. 1** Illustration of altitude variations in the magnetic field associated with an ionospheric current system. Left: Three-dimensional current system from the Laundal et al. (2018) model, showing the radial (field-aligned) current in color, and the horizontal sheet current density at 110 km altitude as vectors. Contours indicate the equivalent current function. Right: Magnetic field disturbances along the dawn-dusk meridian evaluated at four different altitudes. Dashed lines show the field that would be measured by ground magnetometers distributed along the dawn-dusk track in the polar plot. Colored lines represent the magnetic field at 220 km (same distance to the current as the ground, but above), 500 km, and 1000 km</figcaption></figure>
principle forms the basis of the Kamide–Richmond–Matsushita (KRM) technique (Kamide et al. 1981), a precursor to AMIE that derives ionospheric electrodynamics from ground-based magnetic field observations (Friis-Christensen et al. 1985).

Conversely, magnetic field observations from high-altitude spacecraft only detect the FACs, which serve as the source term for the divergent part of **J**. Taking the divergence of Equation (2) and applying current continuity yields an equation that eliminates the components of **J** undetectable by high-altitude satellites. This equation underpins magnetosphere–ionosphere coupling in magnetospheric simulations, where the electric field is determined from a prescribed set of FACs and conductances (Merkin and Lyon 2010). It has also been applied in data-driven studies (Anderson et al. 2016; Robinson et al. 2021; Chartier et al. 2022, 2023) to derive ionospheric electrodynamics maps from global estimates of FACs (see Sect. 2.2.3) and conductances.

Satellites in lower orbits, however, can detect both the curl-free and divergence-free components of **J**. In practice, relating these measurements to **E** requires evaluating both the curl and the divergence of the current as represented by the height-integrated ionospheric Ohm's law (Laundal et al. 2016).

### 2.1.2 AMIE and Lompe

As previously noted, the AMIE and Lompe techniques share similar physical assumptions but differ in their implementation. AMIE employs spherical cap harmonics (SCHs) (Richmond and Kamide 1988; Torta 2020) to model the potential electric field throughout the entire polar region, extending to a boundary typically at 50° magnetic latitude, where the zonal component of the electric field is constrained to zero. This makes AMIE inherently global, with its smallest resolvable spatial scale determined by the truncation level of the harmonics. The implementation described in Lu (2017) achieves a spatial resolution of approximately 200 km in the north–south direction and 500 km in the east–west direction. AMIE uses the background mean and covariance as a prior in its estimation of SCH coefficients (as in the maximum a posteriori estimate). The use of an empirical ionospheric convection model as the background mean helps to regularize its inversion process, thereby constraining the solution in regions with sparse data (see Richmond and Baumjohann (1984) for the background covariance used in AMIE). The background covariances are later updated as part of the AMGeO tool (Matsuo 2020) with principal component analyses of large volumes of SuperDARN plasma drift, DMSP auroral particle precipitation, and Iridium® magnetic field data (e.g., Cousins et al. (2013, 2015); McGranaghan et al. (2015); Shi et al. (2020)). See Matsuo et al. (2005) for the maximum likelihood estimation of covariance parameters and low-rank covariance models based on principal components.

In contrast, Lompe represents the potential electric field as a superposition of spherical elementary current systems (Amm 1997; Vanhamäki and Juusola 2020). These localized basis functions represent line charge densities along radial lines extending from the ionosphere to infinity (Reistad et al. 2019). Though defined globally, their influence is short-ranged, concentrated near their respective poles. The systems are strategically placed to achieve the desired spatial resolution. Lompe arranges these functions on a regular grid, enabling finite differencing to compute the curl and divergence of the ionospheric Ohm's law, a critical step for relating magnetic field perturbations to the electric field. Finite differencing is also employed in the regularization process, to balance model norm minimization with control over spatial charge density gradients.
Thus, AMIE provides a global view, while Lompe enables flexible, regional analyses
tailored to the application and the available data. In both techniques, the true spatial reso-
lution depends not only on grid resolution or SCH truncation level, but also on the data
distribution and regularization. This results in non-uniform resolution that can be difficult
to quantify. Madelaire et al. (2023) proposed a method to quantify spatial resolution in
models that use spherical elementary current systems on regular grids. Their technique
was developed for the Electrojet Zeeman Imaging Explorer (EZIE) satellite mission, but
it is also applicable to Lompe, which employs a similar grid structure and regularization
approach (Laundal et al. 2021). The resolution of AMIE also depends on the data distribu-
tion, but, unlike Lompe, the technique lacks a comparable resolution metric.

Combining datasets with different spatial resolutions can be challenging. Ozturk et al. (2020) presented a technique to integrate high-resolution regional convection data with a global empirical model, which could enable the combination of methods like AMIE and Lompe.

As described in Matsuo et al. (2015), the FAC can be estimated directly from space-
based magnetometer observations without the use of ionospheric Ohm’s law, circumvent-
ing reliance on the conductance assumption in the inversion of magnetometer data. This is
implemented in the AMGeO for inverse analysis of Iridium® magnetic field observations
(Matsuo 2020) as a step toward the optimization of conductance as an integral part of the
inversion as demonstrated for SuperDARN plasma drift and Iridium® magnetic field obser-
vations (Cousins et al. 2015).

## 2.2 Commonly Used Datasets

Here, we briefly review some of the most important datasets that are used in ionospheric
data assimilation, along with others that may be integrated into future methodologies. We
focus on datasets offering large spatial coverage and relatively high temporal resolution,
suitable for operational use (AMGeO Collaboration 2019). Nevertheless, we acknowledge
the value of high-quality measurements by instruments such as incoherent scatter radars
(ISRs) and single satellites, which serve as crucial inputs during special events or for vali-
dation purposes (e.g., Laundal et al. (2022)).

2.2.1 SuperDARN

The Super Dual Auroral Radar Network (SuperDARN) is an international consortium
operating over 40 high-frequency (HF) radars, primarily encircling the auroral zones and
polar caps of both hemispheres (Greenwald et al. 1995; Chisham et al. 2007). The net-
work has recently expanded to mid-latitudes (Nishitani et al. 2019), improving global cov-
erage, but some regions still lack radars due to geographical constraints and geopolitical
factors. SuperDARN measurements of HF radar backscatter, which can be related to ion
drift velocity in the ionosphere (Hanuise et al. 1981; Villain et al. 1985), enable the deriva-
tion of the horizontal electric field. HF radars with large fields of view, covering around
50° in azimuth and more than 3500 km in radial distance, experience significant varia-
tions in wave propagation conditions in their coverage area. Therefore, echoes often do not
cover the entire sounding region and can sometimes be very sparse. Furthermore, only the
line-of-sight component of the ion velocity is measured, requiring assimilation with other
data and models for full velocity vector reconstruction. Temporal resolutions of one minute
can be achieved, but the local time coverage of the radar changes over time, complicating
the continuous monitoring of specific magnetosphere regions. Finally, uncertainties in the altitude of radar echoes can introduce geopositioning errors unless accurately corrected through calibration (Chisham et al. 2021).

### 2.2.2 Ground Magnetometers

Ground-based magnetometers have long been used to indirectly probe ionospheric current systems. At stable geographic locations, magnetic observatories host magnetometer sensors that provide high-quality, long-term, absolute magnetic field measurements. The International Real-time Magnetic Observatory Network (INTERMAGNET) (Kerridge 2001) ensures data quality from its member observatories and facilitates free data exchange between nations. Datasets from this network offer temporal resolutions from minutes to seconds, with many available in near real time. The International Monitor for Auroral Geomagnetic Effects (IMAGE) consists of 57 magnetometer stations and has been established to study auroral electrojets and ionospheric current systems at high latitudes (Tanskanen 2009). Furthermore, SuperMAG provides easy access to magnetic field data to scientists, teachers, and students (Gjerloev 2009).

Magnetic field measurements capture a superposition of various sources, including the Earth's main magnetic field, induction effects, and magnetospheric and ionospheric current systems (Chulliat et al. 2017; Gjerloev 2012). To isolate the source of interest, its signal must be separated from these measurements (Haberle et al. 2024; Van De Kamp 2013; Gjerloev 2012). Tools such as Spherical Elementary Current Systems (SECS) (Amm 1997) enable equivalent current systems to be modeled from measurements.

### 2.2.3 Satellite Magnetometers

Over the past few decades, high-quality magnetic field measurements from low-Earth orbit have been provided by the Danish Ørsted satellite (Olsen et al. 2014), the German CHAMP satellite (Reigber et al. 2002), and the ongoing ESA Swarm mission (Friis-Christensen et al. 2006). These measurements are crucial not only for developing precise models of Earth's main magnetic field but also for understanding ionospheric currents (He et al. 2012; Laundal et al. 2018; Lühr and Kervalishvili 2021). While the high-quality data from these satellites offer valuable baselines in estimates of two-dimensional maps of ionospheric electrodynamics, their measurements are confined to their (one-dimensional) orbital tracks and must be supplemented with data from other sources to derive maps.

In recent years, the number of satellites in low-Earth orbit has surged, many equipped with platform magnetometers designed primarily for attitude control. However, with proper calibration, some of these instruments can provide data suitable for scientific applications (Stolle et al. 2021; Kloss et al. 2021; Olsen et al. 2020). Since 2010, the AMPERE project has used magnetometer data from the Iridium® satellite fleet to produce maps of FACs every two minutes (Anderson et al. 2000; Waters et al. 2020), using a rolling 10-minute data window. Additionally, AMPERE releases raw magnetic field perturbation measurement data, which can be incorporated into techniques such as AMIE and Lompe. This has enabled routine, data-driven estimates of large-scale FACs, offering a significant new dataset for constraining models of ionospheric electrodynamics. Looking ahead, we anticipate that more magnetometer data will become available in the coming years, driven by new techniques for recalibrating existing platform magnetometer datasets (e.g., Styp-Rekowski
et al. (2025)) and the launch of new missions equipped with science-grade instruments (Zhang et al. 2023; Hulot et al. 2024).

### 2.2.4 Auroral Imaging

The data sources mentioned above provide information on electromagnetic fields. However, applying the ionospheric Ohm's law (Equation (2)) in data assimilation also requires information about the ionospheric conductance. Currently, auroral imaging is the only practical method for obtaining accurate, real-time two-dimensional maps of ionospheric conductance. By analyzing multi-spectral auroral images, we can estimate the energy and energy flux of precipitating electrons, which in turn allow us to calculate the ionization rate. The square root of this rate is proportional to the conductance under specific conditions (see Sect. 3).

Alternative techniques exist for estimating ionospheric conductance. For example, plasma density profiles measured by incoherent scatter radars can be used to infer local conductivities (Cai et al. 2013), and instruments aboard low-Earth orbit satellites can directly measure particle precipitation. However, while these methods provide precise conductance estimates at specific points, they cannot capture the two-dimensional spatial patterns needed in ionospheric data assimilation.

Global auroral imaging was last achieved by the Polar (Torr et al. 1995) and IMAGE (Mende et al. 2000) satellites, launched in 1996 and 2000, respectively, which provided large-scale views of the aurora for a few years. Since then, the most reliable source of multi-spectral optical observations from space has been the SSUSI instrument (Paxton et al. 2017) on the DMSP satellites. However, SSUSI scans only a strip along the satellite tracks, meaning their images are neither global nor instantaneous; they take several minutes to acquire and are updated only once every orbit (approximately 90 minutes).

Ground-based cameras also offer a means to infer conductance, but their reliance on darkness and clear skies limits their use to specific regions and time periods (Spanswick et al. 2024).

In Sect. 3.1, we discuss how auroral imaging is used to infer conductance, and in Sect. 3.2, we discuss how these data can be integrated with magnetic field and plasma velocity measurements to co-estimate fields and conductances in future ionospheric data assimilation methods.

### 2.2.5 Total Electron Content and F-Region Plasma Density

The F-region contains the largest fraction of the ionospheric plasma and is therefore the main source of Total Electron Content (TEC) measured along satellite-ground paths. Although TEC is not directly sensitive to the E-region conductances that govern ionospheric electrodynamics, its ubiquity and temporal continuity make it potentially valuable for constraining or validating global models. Section 5.1 discusses the physical framework for this connection; here we outline how TEC is inferred, its main limitations, and how it can complement other data sources.

The F-region of the ionosphere, located at high altitudes, is characterized by long-lived plasma density structures created by the interplay between plasma production/loss and transport, and their persistence reflects the low recombination rates at those altitudes. Directly measuring plasma production and recombination in this region is challenging. However, the F-region is the dominant contributor to the Total Electron Content (TEC).
TEC quantifies the total number of electrons along a path through the ionosphere and can provide valuable information on the temporal evolution of ionospheric convection patterns. The slant TEC (sTEC) represents the electron content along a specific signal path from a satellite to a receiver, integrated along that path.

Multi-frequency radio signals from satellites, such as those from the Global Navigation Satellite System (GNSS), make it possible to determine the TEC along the signal path from the frequency-dependent phase delay introduced by the ionosphere (e.g., Mannucci et al. 1998). The resulting sTEC represents the integrated electron density between a satellite and a ground receiver and can be converted to vertical TEC (vTEC) through geometric mapping functions. In addition to refraction, diffraction introduces stochastic, frequency-independent phase fluctuations that add noise to the signal (Keskinen and Ossakow 1983; Kintner and Seyler 1985; Carrano et al. 2013; McCaffrey and Jayachandran 2019; Zheng et al. 2022). Such effects are typically mitigated through detrending techniques before TEC values are used in analysis (Ghobadi et al. 2020; McCaffrey and Jayachandran 2019; Zheng et al. 2022).

There is an abundance of measurements, particularly from GNSS receivers, from which the plasma content of the F-region can be inferred. However, these data have not been widely used to study ionospheric electrodynamics. Even low-cost GNSS receivers, such as those in modern smartphones, can track multiple frequencies and provide raw data (e.g., GNSS time, pseudoranges, and carrier phase measurements) for TEC estimation (Tomaštík et al. 2024). Large GNSS receiver networks, such as the International GNSS Service (IGS), enable global TEC analysis.

While TEC provides valuable information on the overall plasma content, it is a vertically integrated quantity and does not by itself capture the full spatial complexity of ionospheric electrodynamics. Complementary measurements, such as local electron density profiles from ionosondes (e.g., Reinisch and Galkin 2011), help constrain or validate the vertical plasma density structure. Combining global TEC maps with regional ionosonde data has been explored in ionospheric tomography and data assimilation (e.g., Prol et al. 2021; Ssessanga et al. 2021).

Recent developments in GNSS-based ionospheric tomography provide an additional path toward three-dimensional electron density reconstruction. Such techniques combine slant TEC observations from multiple satellites and receivers to infer volumetric electron density distributions (e.g., Seemala et al. 2014; Hoque and Jakowski 2011). Although standard GNSS tomography is mostly influenced by the F-region, extensions that incorporate complementary data sources, such as ionosondes or low-Earth orbit radio occultation measurements, can help constrain the lower ionosphere. If these approaches could resolve plasma densities in the E-region, it would enable estimation of height-varying conductivities (see Sect. 3). In practice, such applications remain challenging due to sparse ray coverage and the need for strong a priori constraints, but they highlight a potential means of improving future conductance estimates.

The potential role of F-region plasma density and TEC in constraining ionospheric electrodynamics is discussed further in Sect. 5.1.

## 2.3 The Role of Background Models

High-latitude electrodynamics has often been explored through various parameters taken separately, mainly due to the fact that only partial measurements were available by satellite-based or ground-based experiments. However, in regions with sparse or uneven
observations, empirical models provide critical constraints to guide the inversion process. These models relate electrodynamic quantities through specific assumptions or empirical relations. For instance, such relations include inferring electric potential from ion drift velocity, conductivity from the energy and flux of auroral precipitation, and FACs from magnetic perturbations. Integrating these models with advanced techniques helps bridge observational gaps and more accurately capture dynamic processes.

Despite these strengths, empirical models face significant challenges. Data scarcity limits spatiotemporal coverage of the polar caps and auroral zones. This requires building statistical models from data accumulated over several years and classified by conditions such as hemisphere, season, and interplanetary conditions. However, datasets often lack data for certain conditions, necessitating, for example, the reconstruction of electrodynamic parameters in the southern hemisphere from measurements in the northern hemisphere (see discussion by, e.g., Hatch et al. (2024)). As a result, these models often struggle to follow sudden temporal variations or mesoscale structures.

To address the challenges of conventional statistical methods, recent advances in machine learning (ML) and deep learning (DL) offer promising avenues for improvement. For example, improved classification algorithms more accurately separate F-region ionospheric echoes from other sources in SuperDARN HF radar data (Kunduri et al. 2022; Kong et al. 2024). Additionally, neural networks, random forests, and decision trees can provide smoother and more accurate representations of transition periods, such as those related to interplanetary magnetic field (IMF) rotations. Examples include FACs derived from AMPERE data (Kunduri et al. 2020), convection derived from SuperDARN data (Bristow et al. 2022), and auroral precipitation derived from DMSP satellite data (McGranaghan et al. 2021). These techniques can also offer better descriptions of mesoscale structures compared to conventional models (Bristow et al. 2022). Moreover, ML/DL methodologies can refine model input data (e.g. Gowtam et al. (2024)) and emulate the high-resolution outputs of physics-based models. These advances may eventually enable the computation of numerous model ensembles for future data assimilation-based forecasts (e.g., Kataoka et al. (2024)).

### 2.3.1 Electric Field Models

The electric field implied by ionospheric convection in the F-region and above has been extensively studied, resulting in the development of numerous models based on data from satellites, incoherent scatter radars, and coherent scatter radars. Several models rely on vector electric field or line-of-sight ion drift measurements from instruments on low-altitude satellites. These models include those in Heppner and Maynard (1987), Papitashvili and Rich (2002), Hatch et al. (2024) and Weimer (1995, 2005), which are based on data from missions such as Dynamics Explorer 2 (DE-2), the Defense Meteorological Satellite Programme (DMSP), and Swarm. Furthermore, the model in Haaland et al. (2007) is based on high-altitude data from the Electron Drift Instrument on the Cluster satellite constellation. Ground-based models include those in Senior (1990) and Zhang et al. (2007), which draw on data from incoherent scatter radars such as EISCAT, Millstone Hill, and Sondrestrom, as well as those in Ruohoniemi and Greenwald (1996, 2005), Cousins and Shepherd (2010), and Thomas and Shepherd (2018), which use coherent scatter measurements from SuperDARN HF radars.

Despite their diverse origins, the convection patterns obtained from these models are remarkably consistent. However, they often display significant variations in potential
amplitude and in the response of the low-latitude boundary during periods of enhanced geomagnetic activity. These discrepancies often arise from the type of dataset used (satellite-based or ground-based) and the data collection period. For example, models tied to a specific solar cycle, such as the one in Thomas and Shepherd (2018) based on solar cycle 24, may not accurately represent other solar cycles, even with additional data assimilation. In particular, the Weimer model consistently shows higher cross-polar cap potentials (between 5% and 35%; see, e.g., Figure 5 in Hatch et al. (2024)) than those of other models (Haaland et al. 2007; Orr et al. 2023). These differences can lead to significant variations when used as input for other models, such as when modeling the plasmapause position (Pierrard et al. 2008).

### 2.3.2 Field-Aligned Current Models

FACs are most directly derived from space vector field magnetometer data from missions such as Viking, Magsat, ørsted, CHAMP, and Swarm. These measurements suffer from the same spatiotemporal limitations as convection models, such as limited satellite coverage. Well-known models include Iijima and Potemra (1976), with refinements in Iijima et al. (1984); Erlandson et al. (1988); Stauning (2002), which present the dependence of the FAC patterns on interplanetary magnetic field conditions, especially on the dayside and during northward conditions. More recently, advanced global models have been developed using the Swarm three-satellite mission (e.g., He et al. (2012); Laundal et al. (2018); Edwards et al. (2020)). In parallel, the Iridium® commercial constellation delivers less precise but more numerous magnetometer measurements, offering good coverage of the polar caps. The AMPERE project yields near-real-time FAC maps by combining data from multiple Iridium® satellites (Anderson et al. 2008, 2014; Waters et al. 2020).

### 2.3.3 Auroral Precipitation Models

Several auroral precipitation models exist, primarily based on satellite particle detector data. A widely used model, the OVATION-Prime model, relies on observations from DMSP satellites and provides precipitation population data for different seasons and IMF conditions (Sotirelis and Newell 2000; Newell et al. 2009). More recent alternatives have also been developed (Zhu et al. 2021). However, deriving ionospheric conductivities or conductances from these models is challenging, as we discuss in more detail in the next section.

## 3 Estimating Conductance

Ionospheric conductivity, and its height-integrated counterpart, the conductance, relate ionospheric currents or magnetic field perturbations (Δ**B**) to the electric field (**E**) or plasma velocity (**v**), as expressed in Equations (1) and (2). To infer **E** from Δ**B**, or vice versa, we must know the conductance. As we will see below, the conductance is assumed to be primarily determined by the ionization rate, driven by solar extreme ultraviolet (EUV) radiation and auroral particle precipitation. In Sect. 3.1, we briefly review principles of existing methods for estimating conductance from auroral measurements. In techniques like AMIE and Lompe, the conductance is typically a fixed input, making the relationship between current (**J**) and electric potential (**Φ**) linear. Alternatively, conductance could be a model
parameter co-estimated with the electric field in data assimilation techniques, as discussed in Sect. 3.2.

The conductivities in the ionospheric Ohm's law (Equation (1)) depend on plasma density and the ion-neutral momentum transfer collision frequency (Brekke 2013). This frequency, in turn, depends on neutral density, ion and neutral temperatures, and ion mass (Schunk and Nagy 2009). These parameters are typically obtained from empirical models of the atmosphere (Emmert et al. 2021) and ionosphere (Bilitza 2018). Thus, with these parameters fixed, conductivity depends solely on plasma density $n$.

Plasma density $n$ evolves according to the continuity equation

$$ \frac{\partial n}{\partial t} = -\nabla \cdot (n\mathbf{v}) + P - \alpha n^2, \qquad (4) $$

where $P$ is the plasma production rate from ionization, $\alpha n^2$ is the loss rate due to electron-ion recombination and $\alpha$ is the electron-ion recombination coefficient. In the lower E-region (approximately 100–150 km altitude), the recombination rate far exceeds the plasma transport term $-\nabla \cdot (n\mathbf{v})$, rapidly establishing a steady state, where $n \approx \sqrt{P/\alpha}$. The E-region dominates the height integral used to derive Equation (2) from the 3D ionospheric Ohm's law. Thus, conductance is roughly proportional to the square root of the height-integrated plasma production rate, assuming a constant $\alpha$.

At high latitudes, ionospheric ionization is primarily driven by solar EUV radiation and auroral particle precipitation. Since EUV radiation is absent at night and auroral precipitation is, as mentioned, typically confined to higher latitudes, one would expect low nighttime conductivity at mid to low latitudes. However, radar measurements on the nightside reveal nonzero conductivities (Harper and Walker 1977), indicating a background conductivity (Ridley et al. 2004) which is often termed “starlight conductivity.” This term is misleading, as this conductivity likely stems from low fluxes of precipitating particle and from solar EUV radiation scattered by the geocorona, not primarily starlight (Kelley and Heelis 1989).

Considering these three sources of ionization, the height-integrated conductivity is

$$ \Sigma = \int \sigma dz \propto \sqrt{P_{\text{EUV}} + P_{\text{aurora}} + P_{\text{starlight}}}. \qquad (5) $$

Each plasma production process (EUV, aurora, and starlight) contributes independently to the height-integrated plasma production rate $P$. The quadratic addition stems from the $n^2$-dependent loss term in Equation (4) (Robinson et al. 2021). This leads to a similar decomposition of the conductance in terms of source:

$$ \Sigma = \sqrt{\Sigma_{\text{EUV}}^2 + \Sigma_{\text{aurora}}^2 + \Sigma_{\text{starlight}}^2}. \qquad (6) $$

While EUV radiation is relatively stable and predictable, auroral ionization is highly variable and difficult to forecast. We address the challenges of its estimation in the next section.

## 3.1 Estimating Conductance from Measurements of the Aurora

Since auroral ionization can greatly exceed ionization from solar EUV radiation locally, significant effort has been devoted to modeling, quantifying, and predicting its effects. Conductance is largely determined by plasma density, which is governed by ionization
rates. Therefore, auroral conductance is often calculated from particle precipitation char-
acteristics. Ideally, first-principles models would be used to reconstruct density profiles
from the energy spectrum of precipitating particles and atmospheric profiles. However,
this approach requires more information on particle precipitation than is usually available.
Instead, a parameterization in terms of the energy flux and the average energy is used. Rob-
inson et al. (1987) derived equations, known as “Robinson formulae,” which are widely
used to relate energy flux and average energy to Hall and Pedersen conductances. These
equations are derived from simulations that assume a Maxwellian energy distribution for
the incoming particles. More recently, Kaeppler et al. (2023) presented statistically deter-
mined relationships using Poker Flat incoherent scatter radar data. Their results support the
Robinson et al. (1987) formula for Pedersen conductance but suggest that the Hall conduct-
ance tends to be underestimated. In a complementary study, Wang et al. (2024) compared
Robinson et al. (1987) conductances to ISR-derived conductances during close conjunc-
tions between DMSP satellites and EISCAT radars, finding similar behavior and addition-
ally highlighting the need to account for the effect of ion precipitation.

In many cases, the primary source of uncertainty in conductance is not the relationship
between particle precipitation and conductance itself. Instead, it arises from uncertainties
in precipitation properties reflected by the structure and intensity of the aurora, such as
energy flux and average energy. To illustrate this, consider the sequence of 10 global UV
images of the aurora presented in Fig. 2. These were captured by the Wideband Imag-
ing Camera (WIC) onboard the IMAGE satellite (Mende et al. 2000) over three hours on
March 18, 2001. This representative sequence of images demonstrates that precipitation
occurs across a range of spatial scales and can change dramatically in just a few minutes.
No existing empirical model can accurately predict these spatial and temporal variations.

IMAGE also carried additional cameras that, when combined with WIC, enable the esti-
mation of the average precipitation energy (Frey et al. 2003), and, consequently, conduct-
ance. IMAGE went silent two decades ago, and it was the last mission to provide global-
scale images with high time resolution. Since then, ground-based magnetometer and radar
networks have expanded significantly, and the number of low-Earth orbit satellites with
magnetometers has surged. Yet, without global UV auroral imaging, it remains challenging
to connect these diverse observations into a coherent picture (see Sect. 2.2.4). A new satel-
lite mission with such capabilities would not only enhance the scientific value of existing

<figure><img src="image_3.png" /><figcaption>**Fig. 2** Example of the variability in the global aurora, observed by the WIC camera onboard the IMAGE satellite (Mende et al. 2000) from a three-hour period on March 18, 2001, presented as a function of magnetic latitude and local time. WIC observed UV emissions in the 140-190 nm wavelength range</figcaption></figure>
ground networks and satellite constellations but also enable new discoveries, advancing our understanding of the coupled magnetosphere-ionosphere system (see Sect. 6.2).

The images in Fig. 2 have a spatial resolution of approximately 100 km, which is significantly coarser than the typical width of auroral arcs. Given this limitation, observations with all-sky cameras, which can offer higher spatial resolution, are more suitable for studying ionospheric electrodynamics at arc scales (Van Irsel et al. 2024; Lynch et al. 2015; Clayton et al. 2019). Additionally, space-based imaging from low-Earth orbit, such as the SSUSI instrument (see Sect. 2.2.4), offers higher spatial resolution than satellites like IMAGE at the cost of lower temporal resolution. The resolution and spatial coverage required depend on the scientific objective: Global-scale ionospheric electrodynamics require space-based UV imaging, while arc-scale processes demand the fine spatial and temporal resolution of ground-based cameras.

## 3.2 Co-Estimating Conductance and Electrodynamics

If conductances are known, the height-integrated Ohm's law (Equation 2) establishes a linear relationship between currents and electric fields, as discussed in Sect. 2.1.2. However, this assumption is problematic because conductances are typically poorly constrained by data, as discussed above. A more robust approach would treat conductances, currents, and electric fields on equal footing and infer them jointly, given the incomplete and uncertain information about each parameter. In this framework, observations of neutral wind velocity and magnetic field perturbations would constrain both the electric field model (as in AMIE and Lompe) and the conductance. The main challenge is that treating both *E* and conductance as unknowns introduces nonlinearities, necessitating nonlinear estimation methods instead of simpler linear inversion methods. To contextualize this nonlinear framework, we first review past efforts to estimate conductance directly from field observations, where the fields were assumed to be known, before discussing possible approaches to co-estimating conductance alongside other model parameters.

### 3.2.1 Estimating Conductance Directly from Field Observations

Estimating conductance directly from magnetic and electric field observations is challenging, and there have been several attempts to develop a suitable framework for this purpose (Amm 2002; Cousins et al. 2015; Hatch et al. 2024; Weimer and Edwards 2021). By solving Equation (2), Hall ($\Sigma_H$) and Pedersen ($\Sigma_p$) conductances can be expressed in terms of current and electric field in the frame of the height-integrated neutral wind ($E'$) as shown in Equations (9) and (10) in Hatch et al. (2024). These expressions scale with the ratio of the current to electric field magnitudes $J/E'$, so a small misalignment between *J* and $E'$ in regions with weak electric fields can lead to large errors. This issue is compounded by the potentially high uncertainties of electric field measurements and the general lack of data for the neutral wind, which is typically set to zero (see Sect. 4.1). Moreover, with imperfect data, these expressions may yield unphysical negative conductances (Weimer and Edwards 2021).

One way to minimize the issue of misalignment is to use co-located electric and magnetic field data, which ensures spatial and temporal alignment. This approach was pioneered by Sugiura et al. (1982) using Dynamics Explorer 2 satellite data. They assumed that FACs form infinite sheets perpendicular to the satellite's track, closed by Pedersen currents along the track. Under these assumptions, the Pedersen conductance, assumed to be uniform along
the satellite track, emerges as the slope of the along-track electric field versus the cross-track magnetic field perturbation. Olifer et al. (2021) adapted this technique for Swarm data. However, the method's reliance on several strong assumptions limits its reliability, and including additional data would help to relax these constraints.

A more advanced technique, developed by Amm et al. (2015) and evaluated by Juusola et al. (2016), combines electric and magnetic field data to deduce conductances along Swarm satellite trajectories. Hatch et al. (2024) presented another approach, combining the Swarm-based magnetic field model from Laundal et al. (2018) with their own electric field model, derived using a matching parametrization and satellite observations. They also proposed a filtering method to exclude areas where the model results contradict the zero-neutral-wind assumption.

### 3.2.2 Solving the Nonlinear Inverse Problem

As the studies mentioned above show, conductances clearly influence the observed fields, yet this information is not currently used in assimilation techniques like AMIE and Lompe. Co-estimating conductances and the electric field is challenging due to their nonlinear relationship, which likely explains the limited adoption of such techniques in ionospheric physics. Fortunately, inverse theory offers strategies to address such nonlinearities. Here, we briefly overview strategies relevant to conductance and electric field co-estimation.

Methods for tackling nonlinear problems include Markov chain Monte Carlo (MCMC) (Brooks et al. 2011), stochastic optimization, deterministic optimization, state-space estimation, and machine learning. The choice of algorithm, such as simulated annealing (Kirkpatrick et al. 1983) or Hamiltonian Monte Carlo (HMC), depends on the available information and objectives. Simulated annealing, a stochastic optimization method, seeks an approximate solution guided by observations. HMC (Brooks et al. 2011; Betancourt 2018), an MCMC method, leverages both observations and prior distributions to explore the posterior distribution, providing insights not just into the most likely solution but also into the associated uncertainties.

Given the differences in input and output, it is crucial to carefully match the chosen method to the problem at hand. Each method introduces specific requirements for the available data and prior assumptions, and the goal should be to select an approach that aligns with the constraints and objectives of the problem. For example, stochastic methods like simulated annealing are suited for approximating solutions in high-dimensional spaces with fewer assumptions about the system, while MCMC methods allow a more comprehensive exploration of the solution space, especially when prior knowledge can inform the estimation process.

In the context of conductance co-estimation, the goal is to understand and capture the underlying physics. This requires an approach that not only estimates the most likely solution but also characterizes the full range of possibilities, allowing the system's coupling to be explored. Achieving this requires input data that not only includes measurements but also accounts for their uncertainties and covariances to capture the spatial structure of the inputs effectively (Cousins et al. 2013).

## 4 Extending Data Assimilation Techniques to Mid- and Low Latitudes

The AMIE and Lompe techniques are designed for high latitudes, where the magnetic field is nearly radial and neutral winds play a minor role compared to plasma flow. However, these assumptions do not hold at lower latitudes. In Sect. 4.1, we discuss how to better
<figure><img src="image_4.png" /><figcaption>**Fig. 3** Neutral wind vectors (in m/s) at constant pressure levels approximately equivalent to 110, 120, and 130 km altitude poleward of 50°N geographic latitude, with electric potential (in V) contours for April 24, 2023, at 6 UT (IMF $B_z$ ~ 25 nT; $B_y$ ~ -20 nT; Dst~ -220 nT). The TIEGCM simulation is described in Habarulema et al. (2024) and uses data assimilation products from AMIE for high-latitude electric potential and auroral particle precipitation</figcaption></figure>

account for neutral winds in ionospheric data assimilation. Section 4.2 explores replacing the assumption of radial magnetic field lines with a more realistic representation of the magnetic field geometry.

Furthermore, extending these techniques to lower latitudes requires addressing the transition between low and high latitudes. At low latitudes, the hemispheres are coupled by magnetic field lines of the main magnetic field, as described by the IGRF model, whereas at high latitudes, this coupling breaks down (Laundal et al. 2022). This unresolved challenge is also faced by global magnetosphere simulations aiming to model ionospheric electrodynamics as a unified system rather than isolated domains. The complexities of this transition are discussed in Sect. 4.3.

## 4.1 Incorporating Neutral Dynamics

To derive the height-integrated ionospheric Ohm's law in Equation (2), it is assumed that the neutral wind is zero and that the electric field is expressed in the Earth's rotating frame of reference. The zero-neutral-wind assumption is widely adopted due to the scarcity of reliable measurements. Specifically, in the height integration used to transition from the three-dimensional ionospheric Ohm's law in Equation (1) to Equation (2), the neutral wind field at 100–150 km altitude is critical because the conductivities peak within this range. However, most available wind measurements are obtained from above these altitudes, either via in situ satellite observations or F-region emissions detected by Fabry-Pérot interferometers. Since neutral winds vary significantly with altitude toward the lower E-region (Richmond et al. 2003), as illustrated in Fig. 3, these higher-altitude measurements provide limited insight into the neutral wind profile in the lower E-region. Moreover, wind data remains sparse even at F-region altitudes.

Nevertheless, high-altitude neutral wind measurements reveal a strong coupling between plasma and neutral dynamics, contradicting the zero-neutral-wind assumption. For instance, Förster et al. (2008) used in situ cross-track wind observations from the CHAMP satellite to demonstrate that plasma convection significantly influences neutral winds. This finding is supported by comprehensive studies such as Emmert et al. (2008) and Drob et al. (2015), which integrated data from various sources to show this effect. Billett et al. (2018)
used the Horizontal Wind Model (Drob et al. 2015) to demonstrate that ionospheric Joule heating estimates are highly sensitive to neutral wind assumptions. These studies show that the zero-neutral-wind assumption in Equation (2) is driven more by practical limitations than by physical justification.

A promising approach to incorporating neutral winds into ionospheric data assimilation methods, such as AMIE and Lompe, involves coupling these methods with a physical model of atmospheric circulation. In a pioneering study, Lu et al. (1995) used the TIEGCM model (later described in Qian et al. (2014)) to simulate neutral winds influenced by ionospheric electric potentials derived from AMIE data-based maps in the Northern Hemisphere. Joule heating and FACs were subsequently calculated in a post-processing step using these simulated winds. However, this approach lacked two-way coupling, since neutral winds were not incorporated into the AMIE inversion process.

Future data assimilation methods could enhance this approach by integrating neutral winds simulated by a GCM into the inversion process. This would involve using the height-integrated ionospheric Ohm's law in Equation (2) to relate magnetic field observations to the electric field, which is influenced by neutral winds. In this framework, ionospheric electrodynamic data would periodically be provided as input to the GCM. The GCM would then dynamically evolve to generate realistic neutral wind fields, which would be fed back into Equation (2), ensuring a more self-consistent coupling between electrodynamic observations and neutral dynamics.

To illustrate the variability and magnitude of neutral winds at ionospheric altitudes, Fig. 3 presents TIEGCM simulation results of neutral winds at three pressure levels (approximately 110, 120, and 130 km altitude) for April 24, 2023, at 6 UT, with the simultaneous electric potential $Φ$ shown in color. These results demonstrate significant variability in neutral winds across this altitude range, with speeds exceeding 100 m/s. Including these winds into the assimilation process would provide a more realistic relationship between magnetic field perturbations ($\Delta B$) and plasma velocity ($v$), leading to more precise maps of ionospheric electrodynamics. This effect is illustrated in Fig. 4, which shows magnetic field perturbations on the ground (left column) and in space (middle column), together with the ionospheric electric potential (right column) at the beginning of the storm-time recovery phase. The top row is derived from the combined AMIE field-aligned current and wind dynamo contributions, while the bottom row shows the effect of the wind dynamo alone. The figure reveals magnetic field differences up to 50 nT due to the wind dynamo, indicating that neglecting the wind contribution in data assimilation can introduce significant errors.

## 4.2 Accounting for a Non-Radial Main Magnetic Field

At polar latitudes, Earth's main magnetic field is nearly radial, which simplifies the interpretation of both magnetometer data and horizontal plasma velocity measurements. In this section, we discuss how to account for more realistic field geometries when combining these measurements.

### 4.2.1 Accounting for the Horizontal Component of Field-Aligned Currents

Assuming radial field lines, FACs and their associated curl-free horizontal closure currents in the ionosphere do not contribute to the magnetic field below the ionosphere (Fukushima 1971). Under these conditions, ground magnetic perturbations are associated only with the
<figure><img src="image_5.png" /><figcaption>**Fig. 4** Northward component of the magnetic perturbation [nT] at the ground (left column) and at 310 km (middle column) and the electric potential [kV] (right column) for magnetosphere-ionosphere coupling using AMIE field-aligned current and wind dynamo (top row) and wind dynamo alone (bottom row) for April 24, 2024, at 6 UT</figcaption></figure>

divergence-free part of the horizontal current system. This property, which only holds for radial FACs, significantly simplifies the interpretation of ground magnetometer data. It forms the basis of the KRM method (Kamide et al. 1981), as well as the AMIE and Lompe techniques (see Sect. 2.1.2).

Away from the poles, where magnetic field lines are inclined, the magnetic field of FACs becomes detectable on the ground. Here, ground magnetic perturbations are the sum of two components: one that corresponds to the divergence-free part of the ionospheric horizontal current and one that corresponds to the global FAC system, which extends infinitely into space. Applying the standard KRM technique requires separating these two contributions, a challenge not addressed by established techniques to our knowledge.

Engels and Olsen (1998) proposed a method to represent the FAC magnetic field using spherical harmonics. Their method integrates the magnetic field of the divergence-free part of the horizontal component of the FAC, expressed as spherical harmonic expansions, over radii above the ionosphere. We apply this approach in Fig. 5 to demonstrate how assuming radial field lines alters the ground magnetic field and equivalent current compared to a more realistic configuration based on IGRF field lines (see Sect. 2.1.1).

Column A of Fig. 5 shows FACs and a stream function representing the divergence-free currents from the empirical Average Magnetic Field and Polar Current System (AMPS) model (Laundal et al. 2018), for the Northern (top) and Southern (bottom) hemispheres. The AMPS model was run with a solar wind velocity of 350 km/s, IMF $B_z$ of -3 nT, IMF
<figure><img src="image_6.png" /><figcaption>**Fig. 5** Illustration of the effect of the radial field line assumption on the equivalent current. Column A presents an example combination of FACs and the stream function of the divergence-free part of the current component, taken from the AMPS model (Laundal et al. 2018), for the Northern (top) and Southern (bottom) hemispheres. This defines a 3D current system with a corresponding ground magnetic field. Column B displays the radial component of the ground magnetic field (color) assuming FACs follow radial field lines, with black contours indicating the equivalent current. Column C shows the difference in radial magnetic field perturbations at the ground (color) between the assumptions of radial and IGRF field lines. The green contours represent the equivalent current for the IGRF configuration and are plotted with the thin black contours from column B for comparison. Solid and dashed lines indicate positive and negative values of the current functions, respectively. The differences in Column C are exaggerated by a factor of 10 to highlight the impact of field line geometry. Calculations for column C were performed using the method from Engels and Olsen (1998) and code by Skeidsvoll and Laundal (2025).</figcaption></figure>

$B_y$ of 5 nT, F10.7 of 100 sfu, and dipole tilt angle of 18.9°. These values correspond to June 1, 2020, 00:00 UT, used to convert magnetic latitude and local time to geographic coordinates for IGRF evaluations.

The FACs and divergence-free currents in column A give different ground magnetic fields depending on whether the FACs follow radial or IGRF magnetic field lines. Column B presents results for radial field lines, showing the radial component of the magnetic perturbation $\Delta B_r$ at ground level (in color) and the equivalent current, which matches the divergence-free current from column A. In column C, FACs follow IGRF field lines to geosynchronous orbit and then extend radially outward. The difference in ground $\Delta B_r$ between this configuration and column B is shown in color, exaggerated by a factor of 10 relative to column B. The equivalent current for this configuration is shown in green, with the equivalent current of column B overlaid as thin black lines for comparison.

Figure 5 reveals that, at polar latitudes, the two equivalent currents are nearly identical. At lower latitudes, where magnetic perturbations are weaker, the differences become more noticeable. We reiterate that, unlike the equivalent current of column B, the equivalent current shown in green in column C does not precisely match the divergence-free part of the horizontal current, contrary to assumptions in many data assimilation techniques. As demonstrated here, a forward model can map FACs and divergence-free current to the magnetic field, and thus it should be possible to account for this relationship also in the inverse problem. We used spherical harmonics and the integration recipe from Engels and Olsen (1998) here, but other basis functions could be adapted for non-radial field geometries, as
shown by Vanhämaki et al. (2020) for spherical elementary current systems and dipole geometries.

### 4.2.2 Interpreting Plasma Velocity Measurements in the Presence of an Inclined Magnetic Field

Spherical harmonics and spherical elementary current systems are employed in AMIE and Lompe, respectively, to model horizontal electric fields. At higher latitudes, under the assumption of radial magnetic field lines, these representations inherently satisfy the requirement that the electric field is perpendicular to the magnetic field. At lower latitudes, the assumption of a radial magnetic field is inaccurate. Nevertheless, a horizontal electric field representation remains applicable because the missing radial component of the electric field can be determined from the requirement that $E \cdot B = 0$. Thus, the electric field parameterizations developed for high-latitude conditions remain applicable across all latitudes (e.g., Nicolls et al. (2014); Reistad et al. (2024)).

However, at low latitudes it becomes more complicated to use velocity measurements to infer the electric field. Velocity data are often provided along a single horizontal direction. At high latitudes, where the magnetic field is nearly radial, the corresponding electric field component can be computed by rotating the velocity vector 90° and multiplying it by the magnetic field magnitude, as dictated by the generalized Ohm's law for ideal MHD (Equation 3). In the general case, the line-of-sight direction is not horizontal, and the magnetic field is not radial. Consequently, a portion of the measured velocity may represent plasma flow parallel to the magnetic field lines, denoted $v_‖$, which does not contribute to the convection electric field. This parallel flow can be significant and must be taken into account to ensure accurate analysis. For example, Nicolls et al. (2014) developed a technique using the Poker Flat phased array ISR to map the convection electric potential by co-estimating a constant, spatially uniform parallel velocity. In contrast, Tesfaw et al. (2024) proposed a more flexible approach for mapping phased array ISR velocities, treating $v_‖$ as a scalar field that varies across the analysis region.

While these techniques are effective for incoherent scatter radars, applying them to SuperDARN measurements is less straightforward. SuperDARN is typically assumed to measure the Doppler shift of field-aligned density irregularities moving perpendicular to the magnetic field. Under this assumption, a horizontal SuperDARN measurement can be seen to represent the horizontal component of the perpendicular velocity, $v_⊥$. However, this assumption can fail at mid-latitudes, where the mechanisms generating irregularities are not fully understood (see Nishitani et al. (2019), for a review). Deviations from field-aligned irregularities have been observed in phenomena such as sub-auroral ionospheric scatter (Ribeiro et al. 2011, 2012; De Larquier et al. 2014) and in lower-altitude regions near the radar (Milan et al. 2004; Ponomarenko et al. 2016; St.-Maurice and Nishitani 2020). These deviations complicate SuperDARN data interpretation, highlighting the need for more refined models at mid-latitudes.

## 4.3 Removing the Low-Latitude Boundary Conditions

To extend existing ionospheric data assimilation techniques to lower latitudes, it is essential to account not only for the changing magnetic field geometry and the increasing influence of neutral winds, as discussed above, but also for the coupling between the two hemispheres. At high latitudes, ionospheric electrodynamics is primarily
driven by the magnetosphere and can differ significantly between hemispheres. This coupling between the ionosphere and magnetosphere is often understood in terms of FACs. Region-1 (R1) FACs, originating near the magnetopause, are associated with convection flows across the polar cap (Milan et al. 2017). Region-2 (R2) FACs arise within the magnetosphere due to pressure variations in the sunward plasma flow (Tanka 1995). During substorms, the magnetospheric tail current can divert through the ionosphere (Kepko et al. 2015). These changes in FAC configuration may lead to coupling between R1 and R2 currents. This coupling extends beyond high-latitude regions to lower latitudes (Huba and Sazykin 2014), manifesting as penetration electric fields. These fields, though always present, intensify during geomagnetic disturbances, significantly altering plasma dynamics at mid- and low latitudes.

Numerical modeling is often the only practical method to quantify the penetration electric field due to sparse observational data. However, current numerical models struggle to represent the complex, dynamic variations of R1 and R2 currents. Ionospheric electrodynamo solvers typically assume that field lines are equipotential at mid and low latitudes due to high field line conductivities, while prescribing different high-latitude electric potentials for each hemisphere (e.g., Richmond and Maute (2014)). The boundary between the high-latitude and equipotential regions is often crudely approximately as a circle, with its radius determined by solar wind or geomagnetic indices (e.g., Qian et al. (2014); Heppner and Maynard (1987)). The position of this boundary determines the effect of the penetration electric field. An alternative approach uses FACs flowing between the top of the ionosphere and the magnetosphere as the upper boundary, avoiding this sharp boundary and enabling the inclusion of the wind dynamo in the polar region (e.g., Maute et al. (2021)). However, treating FACs as a forcing mechanism (i.e., a current generator rather than a voltage generator) has limitations. To account for interhemispheric potential differences, the electrodynamics must be solved three times: once for symmetric FAC forcing and twice for residual FAC forcing in the two hemispheres. Still, this remains an ad hoc approach, as noted in Maute et al. (2021).

A new approach, building on the electrodynamo framework of Maute and Richmond (2018), proposes solving the electrodynamics from pole to pole, with independent potentials in the polar cap regions of the two hemispheres. At low latitudes, the electric potential remains symmetric as in existing electrodynamics, but between the polar and low latitudes, FAC flow is permitted, enabling weak interhemispheric coupling. This concept offers greater flexibility in modeling R1 and R2 current variations and their connections to low-latitude regions. However, observations are needed to validate that the transition from induced electric fields in the magnetosphere to electrostatic electric fields in the IT models is accurately represented.

Efforts to remove the low-latitude boundary condition, an artificial dividing line between high and low latitudes, primarily aim to improve global simulations of the magnetosphere-ionosphere-thermosphere system (Merkin and Lyon 2010; Ganse et al. 2025). However, this is also necessary to generalize ionospheric data assimilation techniques beyond high latitudes. For example, the AMIE technique prohibits solutions with plasma flow across a low-latitude boundary, a reasonable and useful constraint when targeting high latitudes, where plasma flows are much higher than at mid- and low latitudes. However, future data assimilation techniques that aim to encompass the weaker flows at mid- and low latitudes must use a different approach.
# 5 Time-Dependent Data Assimilation

In current ionospheric data assimilation techniques, measurements are typically collected over a fixed time interval and treated as instantaneous snapshots. Reducing the time window improves the temporal resolution of these snapshots but often decreases the number of available measurements, potentially compromising accuracy. The evolution of ionospheric electrodynamics is inferred by comparing these discrete snapshots. In the previous section, we discussed key physical effects relevant to this snapshot approach that are generally overlooked in existing assimilation techniques. Beyond these steady-state effects, certain processes are inherently time-dependent and require a framework that connects successive snapshots. This section examines some of these dynamic processes in detail and proposes strategies for incorporating time-dependent behavior into future data assimilation methods.

In Sect. 5.1, we explore the assimilation of F-region plasma density data using the continuity equation (Equation 4). The key challenge in assimilating this data arises from the long life time of plasma at higher altitudes. Unlike the E-region density discussed in Sect. 3, where the plasma density $n$ primarily depends on the instantaneous ionization rate $P$, the F-region plasma density depends on additional terms in the continuity equation, such as the term involving the plasma velocity $v$, and their time histories. Consequently, measurements of $n$ provide a historical record of ionospheric electrodynamics that extends beyond individual snapshots.

Another time-dependent process often neglected in conventional data assimilation techniques is induction, which can be separated into ionospheric and ground induction. In the ionosphere, induction, manifested as compressional plasma flows or induction electric fields, is typically neglected to simplify the representation of the electric field **E** to depend only on a single scalar potential $\Phi$. While this approach reduces complexity, incorporating induction could enhance the temporal stability of the solution and allow better use of the time-derivatives of ground magnetometer measurements. This is discussed further in Sect. 5.2.

Ground induction involves telluric currents in the conducting Earth, driven by changes in the ionospheric magnetic field. It significantly alters the 3D geometry of the magnetic field between the ionosphere and the ground, meaning that the observed magnetic field on the ground (and, to a lesser extent, in space) reflects contributions from both internal (telluric) and external (ionospheric) processes. Effective assimilation of these measurements requires isolating the ionospheric signal by correcting for ground induction. Although ground induction is highly dependent on the time history of the ionospheric magnetic field, current techniques often rely on simplifying assumptions to avoid this complexity. This is discussed in more detail in Sect. 5.3.

Finally, in Sect. 5.4 we address the primary driver of temporal variation in high-latitude ionospheric electrodynamics: magnetospheric dynamics. While the ionosphere influences magnetospheric behavior, the magnetosphere is predominantly driven by the solar wind. Even complete knowledge of the ionospheric electrodynamics is insufficient to fully specify the state of the magnetosphere, whereas an accurate magnetospheric model would significantly enhance ionospheric predictions. However, current global magnetosphere models qualitatively capture ionospheric electrodynamics and lack predictive precision. For these models and ionospheric data to mutually benefit, their dynamics must be synchronized. This challenge is explored in Sect. 5.4.
## 5.1 Including TEC/F-Region Plasma Density

In Sect. 2.2.5, we discussed the relative abundance of F-region density measurements. However, using F-region plasma density to directly infer conductance is difficult. As discussed in Sect. 3, the F-region contributes little to the height-integrated conductances described by Equation (6). While F-region plasma density has not been widely used to characterize ionospheric electrodynamics, it is inherently shaped by it (e.g., Fæhn Foll-estad et al. (2019)). Due to its long-lived nature, the spatial structure of F-region plasma at any moment reflects the cumulative effects of plasma production, transport, and decay. This means that F-region plasma density could serve as a rough indicator of past ionization processes that have influenced both the F- and E-regions. However, extracting meaningful conductance information from this historical dependence requires strong assumptions, which introduce significant uncertainty and limits the practical utility of the approach.

The cumulative effects on plasma density $n$ from production, transport, and decay are captured mathematically by the continuity equation (4). At heights where ideal MHD applies, the plasma flows according to $v = E \times B/B^2$, and the continuity equations becomes

$$ \frac{\partial n}{\partial t} \approx -\frac{E \times B}{B^2} \cdot \nabla n + b_x \Sigma_x^2 - \alpha n^2, \qquad (7) $$

where we assume divergence-free plasma flow and that the plasma production at the height in question is proportional to the total height-integrated plasma production $P \propto \Sigma_x^2$, with $x$ denoting either the Hall or Pedersen conductance (See Sect. 3). If the constants $b_x$ and $\alpha$ are known, this equation could relate observations of $n$ to key parameters of the height-integrated Ohm's law in Equation (2): the electric field $E$ and conductances $\Sigma_x$.

To illustrate the time-dependent interplay between plasma density ($n$), electric field ($E$), and production rate ($P$), Fig. 6 presents initial conditions and final results from a numerical integration of Equation (4) over two hours using Euler's method (see also Figure 2 in Hovland et al. (2022)). The simulation uses a fixed $v$ from a static convection pattern from the Hatch et al. (2024) model, a fixed $P$ proportional to a simple solar insolation model, and a recombination coefficient $\alpha$ of $10^{-8} \text{ m}^3/\text{s}$. The initial density is set to $n = \sqrt{P/\alpha}$, which is proportional to the expected conductance pattern (see Sect. 3). During the two-hour simulation, a complex density structure emerges, deviating significantly from the initial pattern. Specifically, a “tongue of ionization” forms as plasma is transported into the polar cap, while a low-latitude trough appears at dusk where plasma motion and production are minimal. Adding time variability to $P$ or $v$ could introduce discontinuities in the tongue of ionization, potentially leading to the formation of polar cap patches (Spicher et al. 2017). Although the tongue of ionization forms in the F-region, it results from the same production processes that also momentarily enhance E-region density and conductance. Because the F-region plasma persists and is transported long after the E-region response has vanished, the resulting structures reflect the time history of plasma production and convection rather than the instantaneous electrodynamics.

Directly incorporating F-region density measurements into data assimilation techniques for ionospheric electrodynamics may not be feasible due to the numerous assumptions required. However, plasma density could serve as an independent validation metric for inversion models. Since plasma density reflects the cumulative effects of plasma transport and production over time, comparing modeled and observed density structures could reveal inconsistencies in the inferred electrodynamic conditions. For example, if a modeled electric field configuration leads to unrealistic plasma density distributions when evolved
<figure><img src="image_7.png" /><figcaption>**Fig. 6** Illustration of the interplay between F-region plasma velocity, production, and decay at altitudes with long plasma lifetimes, such as the F-region. The left panel displays the electric potential (black contours) and velocity (arrows) of a static convection pattern from the Hatch et al. (2024) model, transformed to an inertial frame, overlaid with a simple representation of plasma production by solar EUV radiation (color scale), relative to the subsolar production rate $P_0$. The right panel shows the plasma density after two hours, obtained by integrating Equation (4) from an initial condition matching the production pattern in the left panel, with a recombination coefficient $\alpha$ of $10^{-8}$</figcaption></figure>

through the continuity equation (Equation 7), it may indicate inaccuracies in the inferred
electrodynamic state. This approach could be especially useful in data-sparse regions,
where traditional validation methods that rely on direct electrodynamic measurements are
less suited.

**5.2 Handling Induction Electric Fields in the Ionosphere**

Time-dependent data assimilation is motivated by the need to study the dynamic behavior
of a system. There are two key and complementary reasons why this capability is desirable.

First, analyzing a system’s temporal evolution using a steady-state model, without tem-
poral constraints, can result in sporadic behavior. This issue is particularly pronounced in
ill-posed or ill-conditioned problems, where small variations in data or model parameters
can lead to large fluctuations in the solution. Such instability hinders robust analysis, as it
becomes unclear whether variations in the solution between different times reflect true sys-
tem dynamics or are merely artifacts of an under-constrained solution.

To address the issue of erratic behavior in steady-state models, various temporal stabili-
zation techniques can be applied. One of the simplest approaches is to use the model state
from the previous time as a prior for the current state, effectively penalizing large devia-
tions and promoting temporal coherence. An alternative is to link multiple states across
time using functions that enforce desired properties. For example, splines can be used to
enforce smoothness and allow direct control over temporal derivatives (e.g., Olsen et al.
2006). For example, Ohma et al. (2024) applied temporal splines to a model of auroral
 oval boundaries derived from global images by the IMAGE satellite, effectively elimi-
nating sporadic behavior and enabling calculation of boundary velocities using analyti-
cal methods. This approach also mitigates errors from missing data by interpolating over
observational gaps. Since these linking functions are not constrained by physical principles, their design is flexible, limited only by creativity and problem-specific requirements.

The second motivation for time-dependent data assimilation is to explicitly model the physical mechanisms driving the system’s evolution. In the context of ionospheric electrodynamics, this includes incorporating the inductive electric field to better capture the system’s dynamic response beyond the steady-state approach. More generally, physics-based temporal constraints, such as those derived from differential equations governing the system’s behavior, offer a more robust and interpretable alternative to purely statistical or empirical stabilization techniques. However, these constraints can increase complexity, potentially leading to under-determined systems, increased nonlinearity, and the demand for more sophisticated numerical solvers, which can elevate computational costs.

One well-established approach to enforcing physics-based constraints in data assimilation is Kalman filtering (Kalman 1960). The Kalman filter integrates physical models by incorporating a state evolution equation, which can be derived from fundamental laws such as Maxwell’s equations in electrodynamics. In ionospheric electrodynamics, the total electric field can be decomposed into potential and inductive parts, fundamentally linked through physical laws. However, in discretized model representations, these parts are treated as separate state variables, coupled through the state evolution equation. When the state evolution equation is derived from Faraday’s law, it describes how temporal changes in the magnetic field determine the inductive electric field and also govern the time evolution of the potential electric field, ensuring physical consistency. Building on this, the Kalman smoother extends the approach by considering both past and future states in state estimation, strengthening temporal consistency. By structuring the model in this way, the assimilation framework can enforce physical constraints while maintaining a clear distinction between potential and inductive contributions.

Beyond Kalman filtering, several methods can enforce physical constraints in data assimilation. Variational data assimilation (4D-Var, Dimet and Talagrand (1986)) optimizes a system's trajectory over a time window by minimizing a cost function that incorporates both observations and governing equations. In the context of ionospheric data assimilation, for instance, this approach can ensure that Maxwell's equations and Ohm's law are satisfied across multiple time steps, though it requires an adjoint model and is computationally demanding. Constrained optimization techniques, such as Tikhonov regularization or the use of Lagrange multipliers (Aster et al. 2013), can also penalize deviations from physical laws. These techniques are often used within Kalman filters or variational frameworks to further stabilize solutions.

A more recent development is the use of Physics-Informed Neural Networks (PINNs) (Raissi et al. 2019), which embed governing equations directly into a neural network’s loss function. This allows the model to learn from sparse or noisy data while ensuring compliance with physical laws, such as Faraday’s law. Although promising, PINNs require extensive training data and careful tuning of hyperparameters to effectively balance observational and physical constraints.

Recently, Madelaire et al. (2024) introduced a technique for estimating the ionospheric induction electric field using magnetic field measurements below the ionosphere. Extending this approach to incorporate time dependence could help constrain solutions across time steps, simultaneously reducing the sporadic nature of steady-state solutions and resolving dynamic processes. Their study revealed that local Joule heating can vary by tens of percent (Fig. 7), while the total integrated heating in the model domain only changed marginally. This highlights the importance of capturing time-dependent processes, as steady-state models may underestimate spatially localized induction effects.
Furthermore, the growing focus on mesoscale structures in the ionosphere, exemplified by the recently launched EZIE mission (Laundal et al. 2021; Yee et al. 2021), underscores the need for physical constraints in data assimilation frameworks. The inductive electric field can play a significant role at these scales, particularly on the nightside and in the auro-ral region, where large conductance gradients exist. Accurate modeling of such effects is essential for resolving the dynamics of FACs and understanding energy transfer in the cou-pled magnetosphere-ionosphere system.

## 5.3 Handling Ground Induction

In Sect. 2.1.1, we explained how magnetic field measurements can be used to infer the ionospheric electric field. In particular, we presented the KRM technique (Kamide et al. 1981), which derives the electric field by taking the curl of the ionospheric Ohm's law to isolate the divergence-free part of the horizontal current. The divergence-free part of the current is what ground-based magnetometers primarily measure when assuming radial field lines (see Sect. 4.2). However, this approach presumes that all additional magnetic field sources are accurately accounted for. In reality, a significant portion of ground mag-netic disturbances originate from currents induced inside the Earth by the time-varying ionospheric magnetic field (Marsal et al. 2012). Failing to properly subtract or otherwise account for this induced component introduces a systematic error that can lead to incorrect estimates of the ionospheric electric field.

Accurately accounting for the effects of induced telluric currents is difficult, as the induced magnetic field varies with the frequencies and scale sizes of the inducing field, as well as the 3D structure of the ground conductivity (Fig. 8). However, Gauss (1877)

<figure><img src="image_8.png" /><figcaption>**Fig. 7** Adaptation of Figure 1 from Madelaire et al. (2024). Panel (a) shows the steady-state Joule heating. Panel (b) illustrates the difference when the ionospheric induction electric field is included and also depicts compression flow, i.e., the compression and expansion of magnetic flux necessary to facilitate the transition between two steady states</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Fig. 8** Conductances of subsurface layers in Fennoscandia, including contributions from bedrock, marine sediments, and seawater. Bedrock conductances are given by S-MAP (Korja et al. 2002), while marine sediment and seawater conductances are from Grayver (2021). Contributions from seawater and marine sediments are included in the first panel and assumed negligible at depths greater than 10 km</figcaption></figure>

showed that it is in theory possible to separate external and internal contributions to the magnetic field using magnetometer data alone. This approach, known as the Gauss method, involves modeling two independent current systems, one for ionospheric currents and one for telluric currents, that together generate the observed magnetic field. This approach increases the model complexity significantly, making robust results contingent on extensive magnetometer coverage, which is often lacking. Despite these challenges, recent studies focusing on the Fennoscandia region, where ground magnetometer coverage is relatively dense, have applied the Gauss method with promising results (Juusola et al. 2020, 2023).

An alternative to the Gauss method involves linking induced currents to inducing currents using physical principles. This approach avoids the need to invert for two current systems (internal and external) separately. Another advantage is that it can be applied to satellite data, where ground and ionospheric currents are both internal, and therefore inseparable by the Gauss method (Kloss et al. 2023). Given the conductivity of Earth, the induced currents and their corresponding magnetic field can be calculated from the inducing current using Maxwell's equations and Ohm's law (Grayver et al. 2021). Generally, this calculation requires the full time history of the inducing field, making the model inherently time-dependent. However, some simplifying assumptions are commonly employed. The trivial approach treats the Earth as nonconducting, thereby neglecting induced currents entirely. A better alternative is to assume that there is a superconducting spherical shell at some depth below the surface (Olsen et al. 2005). This latter assumption eliminates frequency dependence, resulting in an instantaneous mapping between the inducing ionospheric currents and the induced (or “mirror”) telluric currents (Juusola et al. 2016).

The mirror current approach, while highly idealized, offers the advantage that it does not increase model complexity and integrates readily with prior information. For instance, when Tikhonov regularization is applied to ionospheric currents in techniques like Lompe, the Gauss method tends to over-attribute the magnetic field to telluric currents (Walker et al. 2023, 2024). Furthermore, refining the Gauss method would require costly expansions of ground magnetometer networks. For these reasons, we advocate for methods that incorporate induction physics rather than relying solely on data-driven separation. In this context, Grayver et al. (2021) introduced a promising approach that employs a realistic ground conductivity model to calculate induced effects from inducing currents and their time history. Although this approach results in an inherently time-dependent model, it is well suited for time-varying current systems.
## 5.4 Connecting Ionospheric Observations and Magnetosphere Simulations

In the ideal data assimilation scenario, observations provide instantaneous information about a system's current state, while a physics-based simulation predicts its future evolution. These two sources of information are continuously integrated to produce a refined state estimate. Approaches for assimilating ionospheric and thermospheric data into upper atmosphere models such as TIE-GCM are discussed in Sect. 4.1. However, for high-latitude ionospheric data assimilation, significant challenges remain. First, the magnetosphere is the primary driver of dynamics in the high-latitude ionosphere, but it remains unclear how effectively, and on what timescales, the magnetospheric state can be modified by integrating ionospheric data into magnetospheric simulations. This issue is exacerbated by the fact that the magnetosphere is largely driven by the solar wind, constantly modifying the ultimate drivers of ionospheric variations. Secondly, even the most sophisticated current magnetospheric models remain limited in their ability to capture observable ionospheric variations with accuracy sufficient for ionospheric data assimilation. This is a function of three primary factors, i.e., incomplete physics, poorly constrained upstream (solar wind) boundary conditions and the inherent high degree of nonlinearity of magnetospheric dynamics. For instance, substorms have profound effects on magnetospheric dynamics and their ionospheric responses but substorm onsets remain unpredictable. Consequently, simulations of the magnetosphere and the observed state of the ionosphere are unlikely to be well-synchronized.

Merkin et al. (2016) made an initial attempt at tackling the issues above. The work was motivated by the goal of assimilating magnetic perturbations measured by Iridium® satellites at ~800 km and aggregated by the AMPERE project. The idea was to modify or “tweak” the magnetospheric source of these magnetic perturbations to optimally match the observations at the Iridium® orbit altitude. In their work, Merkin et al. (2016) focused on the Region 2 FACs that are generated by the divergence of the ring current in the quasi-steady state inner magnetosphere. They used synthetic simulated magnetic perturbation data and constructed a model-derived mapping function between the ring current pressure and the ionosphere. Given an arbitrary distribution of low-altitude magnetic perturbations, the mapping function was used to reconstruct the optimal inner magnetosphere pressure, which was ingested into the simulation, in turn, producing an optimally matching response in the ionosphere. In their work, Merkin et al. (2016) hypothesized that similar techniques could be used to assimilate high-latitude FACs, that are generated at the magnetopause, by running ensemble simulations driven by perturbed upstream conditions, yielding magnetic perturbations in the ionosphere that would optimally match the observations.

Another, rather speculative, approach to informing magnetospheric simulations involves deriving reconnection rates from ionospheric observations of the open-closed boundary and ionospheric convection patterns. The reconnection rate would be inferred from the rate at which magnetic flux crosses the open-closed boundary (Vasyliunas 1984), often assumed to align with the poleward edge of the aurora (Milan et al. 2017). However, this approach has several limitations. First, the auroral boundary may not always accurately represent the open-closed boundary. Second, continuous global monitoring of the auroral oval is beyond current instrumentation capabilities (see Sect. 6.2), and deriving reliable convection maps from available data, whether directly or via data assimilation, is challenging. Finally, since nightside reconnection occurs mainly during substorm events (Milan et al. 2007), simulations would need mechanisms to replicate
such transient phenomena by artificially triggering substorms to match the observed magnetic flux throughput. Recently, Arnold et al. (2023) showcased a related technique, demonstrating the utility of data-informed adjustments to reconnection in global simulations via localized resistivity, albeit on the nightside and using magnetospheric data.

Incorporating ionospheric information into magnetospheric simulations can, in principle, make the coupled system more physically consistent and better synchronized with observations. By constraining magnetospheric parameters such as field-aligned current systems and reconnection rates, ionospheric data can help magnetospheric models more accurately reproduce the conditions that control ionospheric electrodynamics. This feedback thus provides a pathway to more realistic representations of both systems.

# 6 Concluding Remarks

We have presented the fundamental assumptions behind assimilation techniques for polar ionospheric electrodynamics, specifically the AMIE and Lompe techniques, and discussed several ways in which they can be improved:

* *Co-estimation of conductance:* Section 3.2 explains how electromagnetic fields reveal information about conductance overlooked in current techniques and suggests ways to integrate this in future approaches.

* *Incorporating neutral dynamics:* Section 4.1 addresses the common assumption of negligible neutral winds in current techniques and suggests integrating general circulation models into future assimilation techniques for more realistic neutral wind patterns and constraints on the electrodynamics.

* *Accounting for non-radial main magnetic field:* Section 4.2 discusses the widespread assumption that the magnetic field is radial and proposes a method to adopt a more realistic geometry.

* *Removing the low-latitude boundary:* Section 4.3 examines the artificial sharp boundary between high and low latitudes used in existing simulations and assimilation techniques and explores ways to eliminate this constraint in future models.

* *Including F-region density measurements:* Section 5.1 explores how F-region density measurements reflect the history of plasma production and transport and how this could be leveraged in future techniques that account for time dependence.

* *Accounting for induction in the ionosphere:* Section 5.2 addresses the impact of neglecting ionospheric induction and proposes that its inclusion could improve the stability of time-dependent assimilation techniques.

* *Accounting for ground induction:* Section 5.3 discusses the neglect of time history dependence of ground induction in existing assimilation techniques and proposes improvements to account for more realistic ground conductivity structures.

* *Modeling the physics of time variations:* Section 5.4 considers how global MHD simulations could be used in data assimilation techniques for ionospheric electrodynamics.

These points highlight physical processes missing from the models underlying existing data assimilation methods. To conclude this review, we briefly explore additional potential improvements in the techniques themselves and their applications. Section 6.1 examines regularization, an intrinsic component of inversion algorithms. Section 6.2 discusses how data usage can be expanded through new instrumentation and archived data. Section 6.3
considers the operational use of data assimilation techniques for space weather purposes. Section 6.4 discusses the potential of resolving the three-dimensional electrodynamical structures through data assimilation. Finally, Sect. 6.5 discusses how data assimilation methods can be applied to and benefit from Observing System Simulation Experiments (OSSEs).

## 6.1 Regularization

Any ionospheric data assimilation technique is arguably ill-posed and requires the incorporation of physical assumptions. For instance, as discussed in Sect. 5.2, neglecting the inductive electric field in the Ionospheric Ohm's law reduces the number of model parameters and stabilizes the inversion. This approach effectively imposes a hard-coded physical constraint. Alternatively, instead of neglecting the inductive electric field, its magnitude could be constrained through regularization (e.g., Stamm et al. (2023)).

Although regularization has not been the focus of this review, it plays a crucial role in stabilizing inversions, especially as models grow more complex to incorporate more realistic physics. Regularization itself encodes physical assumptions, though often implicitly. For instance, regularization often controls spatial structure or smoothness, but its impact is not always transparent. For example, intuitively understanding the effect of a singular value decomposition (SVD) cutoff, a common regularization parameter, on spherical harmonic coefficient estimates can be challenging, especially when combined with explicit constraints such as spherical harmonic truncation.

In ionospheric data assimilation, techniques such as SVD cutoffs, Tikhonov regularization, and Bayesian priors each impose constraints in distinct ways, yet their physical implications often remain unclear. As model complexity grows, understanding these influences becomes increasingly important. Ultimately, the resulting solutions are a combination of direct measurements, the physical model, and prior assumptions, which may include the use of empirical models. Presenting the balance between these components as transparently as possible is essential. For example, spatial resolution maps by Madelaire et al. (2023) help distinguish regions where the solution is based primarily on data from those where it is highly dependent on prior assumptions. Future studies should prioritize developing and using such metrics to improve transparency.

## 6.2 Future Datasets

The lack of data remains the primary limiting factor in accurately characterizing ionospheric electrodynamics. Specifically, the lack of large-scale, high-temporal-resolution auroral observations restricts our ability to integrate disparate datasets. However, upcom-ing missions may soon address this challenge. For example, the ESA-CAS SMILE mis-sion (Wang et al. 2025) will feature a camera designed to observe the UV aurora glob-ally. ESA's developing Aurora mission, focused on monitoring the aurora for space weather purposes, could enable real-time tracking of ionospheric electrodynamics when combined with the techniques discussed here. Moreover, NASA's selection of MAAX (Liemohn et al. 2023), an auroral imaging mission now in Phase A, signals that the two-decade-long data gap may soon close.

Further advancements are expected from NASA's recently launched EZIE satellites (Yee et al. 2021), which are designed to remotely measure the ionospheric magnetic field and map equivalent currents. If realized, the multi-satellite Geospace Dynamics Constellation
would measure both electric and magnetic fields. These missions will provide invaluable datasets for techniques such as Lompe and AMIE, enhancing our ability to reconstruct ionospheric electrodynamics. Beyond initiatives by national space agencies, the potential for scientific instrumentation on commercial spacecraft, including engineering-type sensors like platform magnetometers (see Sect. 2.2.3), could further expand our observational capacity.

In parallel with these new datasets, revisiting the extensive archive of data collected since the 1970s using modern ML/DL methods represents an exciting opportunity. By reconstructing the ionosphere in 3D over multiple solar cycles as a function of solar wind parameters and magnetic indices, we could significantly improve empirical models. This, in turn, would provide better regularization for data assimilation.

## 6.3 Operational Aspects

The techniques discussed in this paper aim to characterize ionospheric electrodynamics by combining diverse datasets within a unified model framework. In operational space weather settings, these methods have the potential to significantly enhance nowcasting and, eventually, forecasting capabilities. However, transitioning from retrospective analysis to real-time applications introduces several challenges.

Real-time or near-real-time characterization of the ionosphere requires rapid access to data from a broad range of instruments. Currently, data assimilation techniques are primarily used to analyze past events. To move toward operational use, robust infrastructure must be developed to ensure fast data availability. With sufficient data coverage and computational resources, data assimilation can provide timely nowcasts of the ionospheric state. These nowcasts can, in turn, serve as essential inputs for magnetospheric simulations, potentially aligning their outputs more closely with current conditions (Sect. 5.4).

Forecasting, while highly desirable, presents additional challenges. The inherent unpredictability of phenomena such as substorm onsets (Freeman and Morley 2004) limits the accuracy of current magnetospheric models. A key requirement for reliable forecasting is the ability to synchronize magnetospheric simulations with real-time ionospheric observations, as discussed in Sect. 5.4. Achieving this synchronization is a major challenge, as discussed in Sect. 5.4, but it is crucial for producing a more realistic representation of the coupled system.

## 6.4 3D Electrodynamics

This paper primarily focuses on two-dimensional electrodynamics due to the scarcity of 3D data, especially in the critical altitude region of ~100–150 km, where ion-neutral coupling is strongest (Palmroth et al. 2021). A notable exception is the EISCAT 3D phased array incoherent scatter radar system under construction (McCrea et al. 2015). Once operational, its three antenna arrays in northern Norway, Sweden, and Finland will enable routine observations of the ionosphere above the radars, providing 3D plasma density, 3D ion velocity vectors, and ion temperature measurements, at different points within a defined volume.

Reistad et al. (2024) presented a technique to transform point measurements into a continuous 3D representation by fitting spherical elementary current systems arranged in a grid at different altitudes. This can be seen as a step toward generalizing techniques like Lompe and AMIE to three dimensions, as envisioned by Amm et al. (2008). Achieving this
requires accounting for the effect of plasma transport on ionospheric conductivities, since plasma lifetime increases with altitude (see Sects. 3 and 5.1). Modeling conductivities at higher altitudes may require the use of physical models of the ionosphere such as SAMI3 (e.g., Huba et al. (2008); Huba and Joyce (2010), CTiPe (e.g., Millward et al. (2001); Codrescu et al. (2012), or IPIM (e.g., Marchaudon and Blelly (2015, 2020)). These models solve transport equations along magnetic field lines for thermal electrons and major ion species, typically incorporating photoionization and electron precipitation via a coupled kinetic module, which provides access to ionization rates and thermal electron heating rate needed to reconstruct electron and ion density profiles. Assuming that the electron-neutral and ion-neutral collision rates are known, which can be a caveat since collision rates are generally derived from laboratory experiments (see Palmroth et al. (2021); Sarris et al. (2023) for a discussion), it is then possible to reconstruct Pedersen and Hall conductivity profiles along the field lines.

Since simulating conductivities at resolutions suitable for 3D data assimilation can be computationally intensive, 3D grids of ionospheric conductivities for various conditions, such as seasons, solar cycles, or magnetic activity, could be pre-computed and made available to users.

## 6.5 Observing System Simulation Experiments

An Observing System Simulation Experiment (OSSE) is a modeling experiment designed to evaluate the potential impact of new observational systems, typically before their deployment. Such experiments support mission planning by enabling well-informed decisions related to, for example, the value of installing a ground instrument, modifying satellite payloads, or adjusting satellite orbits.

In practice, a physics-based simulation which, in the context of ionospheric data assimilation, would model the magnetosphere-ionosphere system, is combined with an instrument simulation that replicates the measurement process to produce synthetic data. This synthetic data is then used as if it was real observational data in an assimilation technique to derive physical parameters, such as electric fields and ionospheric currents. The derived quantities can then be compared to the simulation's ground truth to evaluate how effectively the instrument would perform with real observations.

For example, Laundal et al. (2021) presented an OSSE using the Gamera simulation (Sorathia et al. 2020) and simulated magnetic field measurements (Yee et al. 2021). Their results indicated that the EZIE mission, if successful, will resolve magnetic field structures with sufficient resolution to distinguish between different proposed scenarios for the sub-storm current wedge (Kepko et al. 2015), thereby achieving the mission's scientific objective. In a similar fashion, recent OSSE-based approaches for next-generation ISR systems such as EISCAT3D (Hatch et al. (2025), and references therein) represent a computationally inexpensive and conceptually simple way to explore the enormous parameter space associated with ISR experiment design.

OSSEs are particularly powerful when combined with data assimilation techniques for ionospheric electrodynamics. Beyond evaluating instrument effectiveness, these experiments can validate assimilation techniques and evaluate the impact of proposed instrument changes, using metrics such as resolution maps (Madelaire et al. 2023). Furthermore, since the goal of data assimilation is to maximize the scientific value of individual measurements, incorporating these techniques into OSSEs can significantly enhance mission planning.
**Acknowledgements** The authors gratefully acknowledge the International Space Science Institute (ISSI) for organizing the workshop on Physical Links Between Weather and Climate in Space and the Lower Atmosphere at ISSI Bern (Switzerland) on January 22–26, 2024. This article is one of the outputs from the workshop.

**Funding** Open access funding provided by University of Bergen (incl Haukeland University Hospital). KML, MM, and ASS were funded by the European Union (ERC, DynaMIT, 101086985).

KML was also funded by the Research Council of Norway through grant 300844/F50.

MD was funded by the ESA PRODEX contract 4000144983.

SMH was funded by the Norwegian Research Council under contract 344061.

VGM and AS were funded by the NASA DRIVE Science Center for Geospace Storms under award 80NSSC22M0163.

AM acknowledges support by NASA awards 80NSSC20K1784 and 80NSSC21K0449, and NASA support of EZIE mission through prime contract 80MSFC20D0004. This research was supported in part by NOAA cooperative agreement NA22OAR4320151.

TM was funded by NSF award AGS-2231409.

**Data and Code Availability** The Python code to make Figs. 1, 2, 5, and 6 is available at https://github.com/DynaMIT-uib/DA_review.

## Declarations

**Conflicts of Interest** The authors declare no conflicts of interest relevant to this study.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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## Authors and Affiliations

Karl M. Laundal¹,² ID · Aurélie Marchaudon³ · Astrid Maute⁴ · Spencer M. Hatch² ·

Florine Enengl⁵ · Tomoko Matsuo⁶ · Margot Decotte² · Michael Madelaire¹ ·

Viacheslav G. Merkin⁷ · Anthony Sciola⁷ · Veronika Haberle⁸ · Andreas S. Skeidsvoll²

<figure><img src="image_11.png" /></figure>

Karl M. Laundal
kalau@space.dtu.dk

Aurélie Marchaudon
aurelie.marchaudon@irap.omp.eu

Astrid Maute
maute@iap-kborn.de

Spencer M. Hatch
spencer.hatch@uib.no

Florine Enengl
florine.enengl@gmail.com

Tomoko Matsuo
tomoko.matsuo@colorado.edu

Margot Decotte
margot.decotte@uib.no

Michael Madelaire
micmad@dtu.dk

Viacheslav G. Merkin
slava.merkin@jhuapl.edu

<figure><img src="image_12.png" /></figure>
Anthony Sciola
anthony.sciola@jhuapl.edu

Veronika Haberle
veronika.haberle@geosphere.at

Andreas S. Skeidsvoll
andreas.skeidsvoll@uib.no

1 Division of Geomagnetism and Geospace, DTU Space, Technical University of Denmark, Centrifugevej 356, 2800 Kongens Lyngby, Denmark

2 Department of Physics and Technology, University of Bergen, Allegaten 55, 5007 Bergen, Norway

3 Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, CNRS, CNES, 9 avenue du Colonel Roche, 31400 Toulouse, France

4 Leibniz-Institut of Atmospheric Physics, University of Rostock, Kühlungsborn, Germany

5 Department of Physics, University of Oslo, P.O. Box, 1048 Blindern, 0179 Oslo, Norway

6 Ann and H.J. Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, 3775 Discovery Drive, Boulder, CO 80303-0429, USA

7 Applied Physics Laboratory, Johns Hopkins University, 11100 Johns Hopkins Road, Laurel, MD 21042, USA

8 Conrad Observatory, GeoSphere Austria, Hohe Warte 38, 1190 Vienna, Vienna, Austria