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#### Integration radius as a parameter separating convergent and divergent spherical harmonic series of topography-implied gravity

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##### Citation

Bucha, B., Kuhn, M. (2023): Integration radius as a parameter separating convergent and divergent spherical harmonic series
of topography-implied gravity, XXVIII General Assembly of the International Union of Geodesy and Geophysics (IUGG) (Berlin
2023).

https://doi.org/10.57757/IUGG23-0423

Cite as: https://gfzpublic.gfz-potsdam.de/pubman/item/item_5016023

##### Abstract

Spectral gravity forward modelling delivers spherical harmonic series of gravitational fields implied by band-limited constant-mass density topographic masses. The potential series converges outside the minimum sphere encompassing the masses but is likely to diverge below the sphere once the masses are realistic enough. Implicitly, masses all around the globe are integrated. To limit the integration to masses inside/outside a spherical cap, a cap-modified spectral technique has been developed recently. In this contribution, we formulate a hypothesis saying that if the integration radius is large enough, then the spherical harmonic expansion of far-zone gravity effects from the cap-modified technique may converge even below the sphere of convergence, say, on the topography. Assuming positive topographic heights, the integration radius needs to be larger than the highest topographic height. Given that an analytical proof seems to be too difficult to find, at least for the authors, we put the hypothesis to a thorough numerical test, in which we forward modelled degree-2160 lunar topographic masses up to degree 10,800. We tested the hypothesis by increasing the integration radius from 2.5 to 100 km (the maximum topographic height is 20 km). Then, the spectrally forward-modelled gravity was validated against a divergence-free spatial-domain integration, obtaining results that are in line with the hypothesis. Still, this is far from a formal proof. Our conclusions could be beneficial for the study of convergence/divergence of spherical harmonics on planetary surfaces and for geoid computations based on spherical harmonic expansion of far-zone gravitational effects.