Setting up an orchestra without a conductor: the algorithm itself decides which instruments (basis functions) and at what volume (weights) should play to reproduce Earth's gravitational symphony.
Two GRACE-FO satellites fly one behind the other at an altitude of 500 kilometers, continuously measuring how the distance between them changes. When a mountain range or an underground water reservoir passes beneath them, the gravitational field shifts slightly, and the satellites either draw closer or drift apart. In December 2022, they transmitted 529,832 such points, roughly one for every 20 kilometers along the orbit. That's like measuring gravity at intervals the length of Manhattan.
Previously, the LRFMP algorithm (Learning Regularized Functional Matching Pursuit) had only been tested on artificial grids of 12,500 points, and it was unclear whether it could handle real-world data volumes. The authors derived a closed-form formula for the inner products of basis functions, replacing slow numerical series. It's as if the orchestra, instead of tuning its instruments each time, received ready-made sheet music that can be played immediately. After 1,600 iterations, the relative data error was about 5%, close to the noise level, while the relative root-mean-square error at the surface was about 17.5%. Errors concentrate in the Andes, the Himalayas, and the Pacific Ring of Fire, where high-frequency details of the field are not visible from such an altitude.
This is the first test of LRFMP on real satellite geometry with half a million points, and it opens the door to practical use in geodesy. The method can be applied to monitor mass redistribution associated with melting glaciers and changes in groundwater storage. The next step is to move from already processed coefficients to raw ranging measurements, which are what the satellites directly record. Then Earth's gravitational symphony will sound even clearer.
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.