Abstract: It is necessary to calculate the direct effect on gravity and the indirect effects on the geoid and quasi-geoid when reducing the topography based on Helmert's second condensation method for solving boundary value problems. The traditional integration algorithms for the near-zone direct and indirect topographic effects arein the double integral form and give some approximate errors when taking the integral kernel at the center of the grid as the average of the grid integral kernel. Furthermore, the traditional integral algorithms for the direct and indirect effects have singularity in the innermost area, which increases the computational complexity. In this paper, the prism formulae of the near-zone direct and indirect effects are deduced, which improves the calculation accuracy on the one hand and simplifies the computation process by removing the singularity in the innermost area on the other hand. To overcome the planar approximation error of the prism models, the curvature of the earth can be taken into account. Finally, experimental results show that the prism algorithm for the topographic effect considering the earth's curvature is recommended when a high-accuracy geoid/quasi-geoid is required.

Key words: near-zone direct effect, near-zone indirect effect, Helmert's second condensation method, innermost-zone singularity, prism algorithms, considering the earth's curvature