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Current satellite missions dedicated to global mapping of the Earth's gravity field are providing accurate global models of the geopotential. Harmonic (Stokes) coefficients of the geopotential derived from satellite observations of its functionals, a potential gradient vector and/or gravity gradient tensor, correspond to the (time-averaged) gravitational potential that is generated by the geoid, topography and atmosphere. The manuscript deals with the effect of static topographical and atmospheric masses on spaceborne observations of the potential gradient vector and gravity gradient tensor that should be applied during their inversion into the geopotential at the geoid level. They would allow for derivation of a harmonic representation of the potential generated only by solid masses inside the geoid and ocean, i.e., harmonicity of the geopotential would apply to the entire space outside the geoid. The geopotential could be then synthesized without problems with diverging harmonic series, i.e., the fundamental condition for application of a truncated harmonic series everywhere outside the geoid would be met. Due to its large numerical values, compensation of topographical masses is outlined using a singlelayer potential. Although not entirely sufficient from a geophysical point of view, this model is often applied in geoid-related computations. Corresponding parameters are evaluated in manners that are consistent with spatial resolution and accuracy of current spaceborne data using spherical harmonic representation of topographical heights and corresponding mass distributions. Theoretical formulations are followed by numerical evaluations.

In the central part of Fennoscandia, the crust is currently rising, because of the delayed response of the viscous mantle to melting of the Late Pleistocene ice sheet. This process, called Glacial Isostatic Adjustment (GIA), causes a negative anomaly in the present-day static gravity field as isostatic equilibrium has not been reached yet. Several studies have tried to use this anomaly as a constraint on models of GIA, but the uncertainty in crustal and upper mantle structures has not been fully taken into account. Therefore, our aim is to revisit this using improved crustal models and compensation techniques. We find that in contrast with other studies, the effect of crustal anomalies on the gravity field cannot be effectively removed, because of uncertainties in the crustal and upper mantle density models. Our second aim is to estimate the effects on geophysical models, which assume isostatic equilibrium, after correcting the observed gravity field with numerical models for GIA. We show that correcting for GIA in geophysical modelling can give changes of several kilometer in the thickness of structural layers of modeled lithosphere, which is a small but significant correction. Correcting the gravity field for GIA prior to assuming isostatic equilibrium and inferring density anomalies might be relevant in other areas with ongoing postglacial rebound such as North America and the polar regions.

This article reviews a spectral forward gravity field modelling method that was initially designed for topographic/isostatic mass reduction of gravity data. The method transforms 3D spherical density models into gravitational potential fields using a spherical harmonic representation. The binomial series approximation in the approach, which is crucial for its computational efficiency, is examined and an error analysis is performed. It is shown that, this method cannot be used for density layers in crustal and upper mantle regions, because it results in large errors in the modelled potential field. Here, a correction is proposed to mitigate this erroneous behaviour. The improved method is benchmarked with a tesseroid gravity field modelling method and is shown to be accurate within ±4 mGal for a layer representing the Moho density interface, which is below other errors in gravity field studies. After the proposed adjustment the method can be used for the global gravity modelling of the complete Earth's density structure.

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