A formula for gravimetric terrain corrections using powers of topographic height: Fid-Bau Portal

A formula for gravimetric terrain corrections using powers of topographic height

Sun, W.

Abstract. The application of Stokes' formula to create geoid undulations requires no masses outside the geoid. However, due to the existence of the topography, terrain corrections must be applied in order to satisfy the theoretical requirement, and refine the geoid determination. A common way to deal with the topography is to expand the inverse of a spatial distance l into spherical harmonics, which contain factors as functions of the mean radius R of the Earth and the topographic height (or the compensation depth) H. Then the factors are expanded into a binomial series, since they can dramatically simplify formula derivations and numerical computations. Terrain corrections are investigated while considering the topographic potential. By introducing auxiliary equations, a general formula for calculating the topographic potential, which is a function of the binomial term m and is expressed in terms of the powers of the topographic height H is derived. This formula can be directly used for calculating the terrain effects in geoid determination and isostatic compensation, by truncating the series according to an accuracy requirement. It can also be used to estimate the effects of each individual power m of the topographic height on terrain corrections. Practical formulas for the approximations are derived for this purpose, assuming a maximum topographic height, which is constant. The numerical results agree with those of current studies. Some related theoretical and numerical problems are also discussed.