Topographic effects by the Stokes-Helmert method of geoid and quasi-geoid determinations

The topographic potential and the direct topographic effect on the geoid ar e presented as surface integrals, and the direct gravity effect is derived as a rigorous surface integral on the unit sphere. By Taylor-expanding the integrals at sea level with respect to topographic elevation (H) the power series of the effects is derived to arbitrary orders. This study is primari ly limited to terms of order H-2. The limitations of the various effects in the frequently used planar approximations are demonstrated. In contrast, i t is shown that the spherical approximation to power H-2 leads to a combine d topographic effect on the geoid (direct plus indirect effect) proportiona l to (H) over tilde(2) (where terms of degrees 0 and 1 are missing) of the order of several metres, while the combined topographic effect on the heigh t anomaly vanishes, implying that current frequent efforts to determine the direct effect to this order are not needed. The last result is in total ag reement with Bjerhammar's method in physical geodesy. It is shown that the most frequently applied remove-restore technique of topographic masses in t he application of Stokes' formula suffers from significant errors both in t he terrain correction C (representing the sum of the direct topographic eff ect on gravity anomaly and the effect of continuing the anomaly to sea leve l) and in the term t (mainly representing the indirect effect on the geoida l or quasi-geoidal height).