Modification of Stokes and Vening-Meinesz formulas for the inner zone of arbitrary shape by minimization of upper bound truncation errors

Traditionally, the evaluation of geoidal height by Stokes formula and the vertical deflection by Vening-Meinesz one, and the estimation of the influence of neglecting the distant zone on computing the geoidal height and the vertical deflection were done by taking the inner zone as a spherical cap. It is not very convenient from the point of view of modern numerical methods such as fast Fourier and Hartley transforms where the inner zone is not a spherical cap, but a spherical trapezoid. So, we generalized the known formulas for evaluating the geoidal height and the vertical deflection for an integration area of arbitrary shape. The corresponding formulas for computing the effects of neglecting the distant zone have been derived. Some issues on computation techniques have been investigated. As an example, the case where the inner zone is modeled as a spherical trapezoid was given special attention, and practical computations were performed.