Gravitational attraction of local crustal masses in spherical coordinates

Abstract

The gravitational attractions of terrestrial masses and condensed terrestrial masses were modeled in local regions of gravity stations in different ways. These differences in the models included the type of coordinate frame (Cartesian versus spherical), grid spacing (30 vs 3 arcseconds), and the shape of the terrain (“flat-topped” vs “sloped-topped” prisms). The effect of each of these variables is quantified for its overall impact on Helmert gravity anomalies. The combined effect of removing the masses and restoring the condensed masses is also compared to classical terrain corrections for suitability in computing Helmert anomalies. Some detailed conclusions are drawn from these test computations. The effect of the Earth's curvature has both a near-field effect (due to the differences in volume and shape between rectangular and spherical prisms) and a far-field effect (due to physical location of masses below the horizon). The near-field effect can achieve 0.4 mGal in the Rocky mountains, and affect the geoid by up to 7.5 cm. Additionally, the approximation of the terrain by flat-topped prisms (even at fine spacings such as 3 arcseconds) is inappropriate for terrain near the station, where errors of 20 mGal have been computed using 30-arcsecond data. It is concluded that when 30-arcsecond terrain is allowed to have a more curved (bilinear) prism top, its gravitational attraction is a significantly closer approximation of 3-arcsecond terrain, even for the prism surrounding the station, as compared to the case of 30-arcsecond flat-topped prisms. It is suggested that classical terrain corrections, for many reasons, should not be used to compute Helmert anomalies. Considering only the accuracy, and not the speed, of the computations, the following conclusions are drawn: terrain effects computed inside a local “cap” should be done exclusively in spherical coordinates with a 3-arcsecond Digital Elevation Model (DEM) out to 0.2∘ radius, and then a 30-arcsecond DEM from 0.2 out to 3.5∘. In all cases, bilinearly shaped prism tops should be used.