Recovering Earth's gravitational field from satellite data is a classic inverse problem known as the downward continuation problem. Alongside this work in the archive are studies using neural networks to analyze gravitational lenses, quantum clocks to detect time dilation, and pulsars to test gravitational waves. All address different aspects of one big question: how to extract gravity information from indirect measurements. This paper stands apart from its neighbors methodologically: instead of machine learning or quantum effects, it employs an iterative greedy algorithm with learning that selects an optimal set of basis functions from a dictionary containing both global spherical harmonics and local radial basis functions.
The LRFMP (Learning Regularized Functional Matching Pursuit) algorithm iteratively builds an approximation of the gravitational potential as a linear combination of dictionary elements. The dictionary includes spherical harmonics (global polynomials), Abel–Poisson kernels (low-pass filters), and Abel–Poisson wavelets (band-pass filters). At each step, the algorithm selects one new element that maximizes an objective function related to the Tikhonov functional. The key improvement for handling large data is deriving a closed form for the inner products of two radial basis functions in the Sobolev space. Previously, these inner products were computed via truncated series using Clenshaw's algorithm, which consumed most of the computation time. Now they are expressed as finite combinations of derivatives of a simple function, dramatically speeding up computations. Experiments were conducted on EGM2008 data (up to degree 2190) and GRACE-FO data (up to degree 96) from December 2022, projected onto the kinematic orbits of the two GRACE-FO satellites. In total, 529,832 points on the orbits were used, and a Driscoll–Healy grid of 1,662,701 points was used for solution evaluation.
After 1600 iterations for EGM2008 data and 1400 for GRACE-FO, the relative data error (RDE) was about 0.05, close to the 5% noise level. The relative root-mean-square error (RRMSE) on Earth's surface reached approximately 17.5% for both datasets. This reflects the severe ill-posedness of the downward continuation problem: even with accurate data, recovering high-frequency details is impossible due to the exponential decay of the operator's singular values. Errors concentrate in regions with strong local structure: the Andes, the Himalayas, and the Pacific Ring of Fire, where short-wavelength components of the field are lost when observed from 500 km altitude. Comparison with previous work shows that this is the first time the method has been scaled to a realistic amount of data: previously, only about 12,500 points on artificial grids were used.
This work demonstrates that LRFMP can handle data volumes comparable to real geodetic problems without losing accuracy. Unlike neighboring studies that rely on neural networks or quantum effects, this approach is purely mathematical with a learnable dictionary. This is the first test of LRFMP on realistic satellite geometry with half a million points, making the method competitive for practical use in geodesy. The flexibility of the dictionary allows combining the advantages of global and local bases, avoiding both the striping noise of spherical harmonics and the overfitting of mascons.
The next logical step is to move from level-2 data (already processed coefficients) to level-1B data or range-rate measurements directly recorded by the satellites. This will require changing the operator T in the mathematical formulation, but as the authors note, now that the method handles large data, such adaptation becomes feasible. A more challenging task remains incorporating a realistic stochastic noise model for GRACE-FO, which is anisotropic and not reducible to simple multiplicative noise. In the long term, the algorithm could be applied to other inverse problems in geophysics, such as recovering subsurface density from gravity anomalies.
The method will directly impact data processing for gravity satellite missions like GRACE-FO and climate change monitoring through mass redistribution observations. It may also prove useful in related fields requiring solutions to ill-posed inverse problems with large data volumes, such as magnetic tomography or seismology.
The authors plan to adapt LRFMP to level-1B data and range-rate measurements, and to investigate methods for choosing the regularization parameter and the minimum number of iterations to achieve comparable accuracy. Additionally, a more realistic noise model needs to be implemented.
An open question in geodesy is how to optimally choose basis functions for representing the gravitational field without a priori assumptions. This work links that problem to the general theory of ill-posed inverse problems, showing that learnable dictionaries can replace fixed function sets. This directly relates to the long-standing debate over spherical harmonics versus mascons: LRFMP offers a compromise that could reduce systematic errors in estimates of mass balance for glaciers and aquifers.
Each GRACE-FO satellite completes approximately 496 orbits around Earth in December 2022, crossing the equator twice per orbit. That gives 1984 points at the equator, corresponding to a resolution of about 20 km: as if you were measuring gravity at intervals the length of Manhattan.
In Arthur C. Clarke's novel 'The Fountains of Paradise', engineers build a space elevator that requires precise knowledge of Earth's gravitational field. The method described here for recovering the potential from satellite data could be part of such technology.