• Bouguer correction;
  • free-air correction;
  • gravity;
  • surface load;
  • vertical displacement


We study the ratio between the gravity variation and vertical displacement at the surface of a self-gravitating spherical elastic earth model when a load is applied at the surface. By expanding both the gravity variation and vertical displacement in series of spherical harmonics, we investigate the spectral behaviour of the ratio of the harmonic components of the gravity variation and vertical displacement. Special attention is paid to the asymptotic limit, that is, for large degrees, and to the mean value for a given degree range, without considering the spectral properties of the load itself. First, we split the gravity variation into two parts: the first term stems from the direct attraction of the load, the second one is due to the elastic deformation of the model. The origin of the second term is twofold: the surface is displaced in the Earth's gravity field and the position of the Earth's mass particles is changed. The ratio between the gravity variation associated to the displacement of the surface and this displacement is the well-known free-air gradient of −0.30 μGal mm−1. We show, by considering different earth models, that the ratio between the harmonic components of the elastic gravity variation (i.e. mass redistribution and free-air effect) and the corresponding components of the vertical displacement tends to the Bouguer corrected gradient of −0.20 μGal mm−1 when both the harmonic degree n of the components is very large and the top layer of the earth model is incompressible. Second, we compute, for each n, the ratio of the components of the gravity variation and those of the vertical displacement omitting the local Newtonian attraction term. Whereas it weakly depends on n for n≲ 6, it is nearly constant for n≳ 6. By defining an average value for those ratios, we obtain a fairly good approximation of the ratio between the total gravity variation and vertical displacement, only valid outside the loaded area, which is −0.26 μGal mm−1. Finally, to check out our spectral result, we consider the loading effect of spherical caps of various angular apertures. We find a mean value of −0.21 μGal mm−1 outside the loaded area which is by 20 per cent in agreement with our spectral result.