Field modelling based on a decade (2014-2023) of combined ground-observatory and low-Earth-orbit satellite magnetic data can now separate the fast-varying geomagnetic field into its ionospheric, magnetospheric, and Earth-induced contributions without prescribing temporal harmonics or a subsurface conductivity model, but the published framework is restricted to mid-latitudes (data between 5 and 56 degrees quasi-dipole latitude, 3-hour time bins, spherical-harmonic degrees up to 5), treats static observatory biases in a separate quiet-night step because joint estimation with a freely time-varying ionosphere is provably non-unique, and interprets the estimated transfer functions with a one-dimensional (radially symmetric) conductivity approximation. The open problem is to extend this source-separation framework to a truly global model that parametrises and co-estimates the polar and equatorial electrojet current systems, achieves a principled and unique treatment of observatory bias, and feeds the separated external and induced coefficient time series into a full three-dimensional (Q-matrix-based) joint inversion for mantle electrical conductivity, with transfer-function stability demonstrated across geomagnetic storms and solar-cycle phases. A 2025 follow-up already pushed the time resolution to one hour and added the Macau Science Satellite-1 data, but kept the mid-latitude restriction and the one-dimensional induction interpretation, so the global and three-dimensional cores remain open.

Background

The geomagnetic field observed at Earth's surface and at satellite altitude varies on all time scales. At periods from hours to a few years, the variations are dominated by three superposed sources: (i) currents flowing in the ionosphere, the electrically conducting layer of the upper atmosphere at roughly 100 km altitude, comprising the solar-quiet (Sq) current system driven by solar heating, the narrow equatorial electrojet that flows along the magnetic equator, and the polar (auroral) electrojets at high latitude; (ii) currents in the magnetosphere, the cavity carved out of the solar wind by Earth's field, including the ring current, magnetotail currents, and field-aligned currents that connect the magnetosphere to the auroral ionosphere; and (iii) induced currents generated inside the electrically conducting solid Earth and oceans by every time variation of the external fields. Separating these three contributions is mathematically underdetermined from either data type alone: ground observatories sit beneath all external sources, while satellites orbit above the ionospheric currents, and only the combination of ground and satellite data makes the separation tractable. The standard parametrisation expands each source's potential field in spherical harmonics whose coefficients (Gauss coefficients) are estimated from the data; following the sources, 'external' coefficients describe ionospheric plus magnetospheric fields and 'internal' coefficients the induced fields.

Min & Grayver (2025) built exactly such a joint ground-satellite model: using roughly 150 ground observatories and six low-orbit magnetometer satellites (Swarm, CryoSat-2, GRACE-FO) over 2014-2023, they estimated piecewise-constant spherical-harmonic coefficients in 3-hour bins (ionosphere to degree 5, induced and magnetospheric fields to degree 4), with no prescribed daily or seasonal harmonics and no assumed subsurface conductivity, and they analysed the resulting time series through storm dynamics and lunar-solar periodicities. Because the induced field is co-estimated rather than modelled from a conductivity profile, the external and internal coefficient time series yield electromagnetic transfer functions: in a radially symmetric (1-D) Earth each internal coefficient is proportional to the corresponding external coefficient through the QQ-response Qn(ω)Q_n(\omega), a complex function of frequency that encodes subsurface conductivity, commonly converted to the CC-response whose real part approximates the sounding depth. For laterally varying (3-D) conductivity this one-to-one relation breaks down: a single external spherical-harmonic mode induces many internal modes, coupled through a matrix of frequency-dependent responses called the Q-matrix.

The same authors documented the framework's limitations, which define the current frontier. First, localised current systems — the equatorial and polar electrojets — cannot be described by low-degree spherical harmonics, so all data outside 5 to 56 degrees absolute quasi-dipole latitude (a coordinate tied to Earth's main-field geometry) are discarded, and a global model requires dedicated parametrisations, for example line loops or point currents. Second, co-estimating static observatory baseline biases together with an unconstrained time-varying ionosphere has an intrinsic nullspace (proven in the paper's appendices): an arbitrary time-invariant field can be exchanged between the ionosphere coefficients and the biases, so uniqueness needs additional constraints; the published workaround estimates biases separately from quiet night-time data with the ionosphere omitted. Third, the transfer functions are interpreted under the 1-D radial approximation, whereas realistic 3-D mantle conductivity couples modes through the full Q-matrix, and a joint inversion of the separated coefficients for 3-D conductivity remains undone — with the practical concern that ionospheric signal leaking into magnetospheric or induced estimates (source effects) would bias any conductivity model built on them, a known problem for earlier satellite-based 3-D conductivity studies that subtracted quiet-time ionospheric models instead. Adjacent work marks the boundaries of what is already achieved: Kruglyakov & Kuvshinov (2025) modelled the equatorial electrojet, polar electrojet, Sq, and magnetospheric sources jointly in the time domain with 3-D induction effects in the forward modelling, but from ground observatory data only, where the ionosphere-magnetosphere separation rests on prior source parametrisations, and with 3-D conductivity imaging stated only as the ultimate goal; Šachl, Knopp & Velímský (2024) performed a satellite-based 3-D conductivity inversion, but for ocean-tide-induced magnetic signals, which sidesteps the external-source separation problem; and Grayver, Min, Olsen & Smith (2025) applied the joint ground-satellite approach to the May and October 2024 geomagnetic storms at 1-hour resolution including the Macau Science Satellite-1 (MSS-1), yet still for mid-latitude sources only.

Problem Statement

Extend the joint ground-observatory and low-Earth-orbit-satellite separation of the fast-varying geomagnetic field so that it becomes a globally valid and uniquely solvable model whose outputs support three-dimensional mantle-conductivity imaging. Concretely, relative to the published decade-scale framework of Min & Grayver (2025) and its 1-hour, MSS-1-enabled storm application by Grayver et al. (2025), the open problem comprises four jointly serving objectives: (a) incorporate dedicated parametrisations of the localised polar (auroral) and equatorial electrojet current systems so that observations at all latitudes can be used instead of being cut at 5 and 56 degrees absolute quasi-dipole latitude, while preserving the separation of ionospheric, magnetospheric, and induced contributions; (b) replace the separate quiet-night bias-estimation step with a co-estimation of static observatory biases and the time-varying ionosphere that is provably unique, or with a principled constrained formulation whose assumptions and remaining nullspace are explicitly characterised, going beyond the intrinsic non-uniqueness proven in the source paper's appendices; (c) use the resulting source-separated external and induced spherical-harmonic coefficient time series in a full three-dimensional inversion for mantle electrical conductivity based on the Q-matrix relation (each external mode inducing a full spectrum of internal modes), replacing the current one-dimensional radial Q-response interpretation, and demonstrate that ionospheric source leakage is controlled; and (d) validate the stability of the derived transfer functions and of any recovered 3-D conductivity structure across many geomagnetic storm events and different solar-cycle phases. Items (a)-(d) serve one overarching objective — a physically complete, globally resolved, source-separated model of the fast-varying field that is trustworthy input for induction studies and space-weather applications; purely mission-dependent coverage improvements (e.g. local-time sampling from NanoMagSat, a planned European Space Agency Scout-constellation magnetometry mission not yet launched, and low-latency data products) are enabling conditions for these objectives, not objectives themselves.

References

  1. Min, Jingtao; Grayver, Alexander (2025). 'A decade of the fast-varying ionospheric and magnetospheric magnetic fields from ground and multisatellite observations.' Geophysical Journal International 241(2):797-825. DOI: 10.1093/gji/ggaf065, arXiv:2412.10601. https://doi.org/10.1093/gji/ggaf065
  2. Grayver, Alexander; Min, Jingtao; Olsen, Nils; Smith, Ashley (2025). 'Assimilation of Ground and Satellite Magnetic Observations Unravels the Ionospheric, Magnetospheric, and Induced Fields During the May and October 2024 Geomagnetic Storms.' Geophysical Research Letters 52(18):e2025GL116964. DOI: 10.1029/2025GL116964. https://doi.org/10.1029/2025GL116964
  3. Kruglyakov, M.; Kuvshinov, A. (2025). 'Joint Time-Domain Modeling Magnetic Field Variations of Ionospheric and Magnetospheric Origin: Conceptual Framework and Practical Implementation.' Journal of Geophysical Research: Space Physics 130(2):e2024JA033328. DOI: 10.1029/2024JA033328. https://doi.org/10.1029/2024JA033328
  4. Yao, Hongbo; Xu, Juyuan; Zhang, Keke (2025). 'A method for simultaneously determining Earth's magnetic field and mantle conductivity models using MSS-1 and Swarm satellite magnetic data.' Physics of the Earth and Planetary Interiors 358:107296. DOI: 10.1016/j.pepi.2024.107296. https://doi.org/10.1016/j.pepi.2024.107296
  5. Shi, Yalei; Lesur, Vincent; Thébault, Erwan (2026). 'Characterizing the magnetic signal generated in the magnetosphere from 1996 to 2024 using ground geomagnetic data.' Earth, Planets and Space 78(1):38. DOI: 10.1186/s40623-025-02363-x. https://doi.org/10.1186/s40623-025-02363-x
  6. Olsen, Nils (2025). 'Accounting for quiet-time magnetospheric field contributions in geomagnetic field modelling.' Physics of the Earth and Planetary Interiors 366:107411. DOI: 10.1016/j.pepi.2025.107411. https://doi.org/10.1016/j.pepi.2025.107411
  7. Šachl, L.; Knopp, O.; Velímský, J. (2024). 'Electrical conductivity of the suboceanic upper mantle constrained by satellite-derived tidal magnetic fields: three-dimensional inversion, validation and interpretation.' Geophysical Journal International 238(3):1254-1268. DOI: 10.1093/gji/ggae209. https://doi.org/10.1093/gji/ggae209
  8. Min, Jingtao; Grayver, Alexander (2023). 'Simultaneous inversion for source field and mantle electrical conductivity using the variable projection approach.' Earth, Planets and Space 75(1):83. DOI: 10.1186/s40623-023-01816-5. https://doi.org/10.1186/s40623-023-01816-5