A Comparison of the classical and recent formulae of handling the effects of close and distant topographic masses in gravimetric geoid computations

Abstract

Different procedures for considering the effects of topographic masses in Stokes- Helmert scheme of geoid determination are reviewed. Classical integral formulae use a planar approximation of the geoid and a limited area of integration, and thus account only for the local contributions of topographic effects. In the other hand, the spherical harmonic representation of the topographic effects normally only includes the longwavelength information. Another description of the effects of topographic masses given by Martinec and Van´ı˘cek (1994a, b) uses near and far-zone integration areas and a spherical approximation of the geoid. Finally, two formulae, that combine short and long-wavelength contributions, are presented for topographical effects. These recent formulae imply that the integral formulae for determining the topographic effects may have some numerical problems in representing global information (for truncated integration domain). On the other hand, a representation of the effects by a set of spherical harmonic coefficients of the topography to, say, degree and order 360 leads to omission of significant short-wavelength information. All above-mentioned procedures were used for computation and comparisons in a test area in Iran with the maximum elevation of 3053 m. The results of these comparisons show that the Martinec and Van´ı˘cek (1994a, b) integral formulae and the recent combined formulae presented by Sj¨oberg and Nahavandchi (1999) and Nahavandchi (2000) are in good agreement with each other. These formulae use a spherical approximation of the geoid, contrary to the classical formulae which use a planar approximation. Only the combined formulae include, however, all wavelength constituents and are recommended for precise geoid determination. Further, the gravimetric geoidal heights were computed applying these different procedures of handling the topographic effects. The results were then compared at Global Positioning System (GPS)-leveling stations in Iran. The standard deviation of the fit with the combined formulae is the best among the other methods and is equal to ±10.1 cm.