Efficient computation of gravitational effects and curvatures for a spherical zonal band discretized using tesseroids

The gravity field modelling due to mass distributions of the Earth is one of the primary fields in geodesy and geophysics. Among the mass bodies, a spherical shell has become a commonly used mass body to evaluate the gravitational effects of a tesseroid due to its analytical solutions and simple formulae. However, this classic numerical strategy has the shortcoming that its computation time is longer with a finer grid size of the discretized tesseroids, and it has to be performed on a high-performance computer. In this contribution, the simpler analytical expressions for the radial gravity vector and radial–radial gravity gradient tensor of a homogeneous spherical cap and spherical zonal band are derived. Moreover, new analytical formulae of the gravitational curvatures (i.e., third-order derivatives of the gravitational potential) of a homogeneous spherical cap and spherical zonal band are also derived. The analytical consistencies between the new and old radial gravity vector and radial-radial gravity gradient tensor of a spherical cap are confirmed. The computation time and relative approximation errors between a spherical zonal band and spherical shell discretized using tesseroids are quantitatively analyzed with different grid sizes. Numerical experiments show that the computation time of a spherical zonal band discretized using tesseroids is about 180/ n times less than that of a spherical shell discretized using tesseroids for the gravitational effects up to gravitational curvatures with different grid sizes both in double and quadruple precision, where n is from the grid size n ∘ × n ∘ . Moreover, the mean values of the relative approximation errors of the gravitational effects of a spherical zonal band discretized using tesseroids are smaller than those of a spherical shell discretized using tesseroids with the influence of the computation point’s height at different grid sizes. Numerical results confirm the benefit of a spherical zonal band in comparison with a spherical shell discretized using tesseroids regarding both the computation time and errors. The numerical strategy of a spherical zonal band discretized using tesseroids can be applied instead of the commonly used numerical strategy of a spherical shell discretized using tesseroids in the numerical evaluation of a tesseroid with different numerical methods in the future research.