Ellipsoidal representation of the topographical potential and its vertical gradient

Abstract.

Due to the Global Positioning System (GPS), points on and above the Earth’s surface are readily given by means of a triplet of the Gauss surface normal coordinates L,B and H called ellipsoidal longitude, ellipsoidal latitude, and ellipsoidal height, respectively. For geodetic applications, these curvilinear coordinates refer to the international reference ellipsoid GRS80, which is an equipotential surface of the Somigliana–Pizzetti reference potential field. Here, we aim at representing the gravitational potential, that is generated by the ‘topographical’ masses above GRS80, and its vertical gradient, i.e. an effect on measured gravity, in terms of the Gauss surface normal coordinates (L,B,H). The spatial (integral) formulas for the topographical potential and its vertical gradient are presented as a sum of a spherical term and corresponding ellipsoidal correction. The formulas for the terrain contribution are evaluated over the test area in the Canadian Rocky Mountains using an area-limited discrete integration. The spectral (series) representation of the topographical potential is also introduced, and the condensation of the topographical masses on or inside the reference ellipsoid is discussed in terms of a simple layer density. The ellipsoidal corrections might seem to be of limited significance in view of a relatively low accuracy of currently available topographical data, especially the mass density. However, the representation of the topographical potential and its vertical gradient using the coordinates that are directly observable with a high level of accuracy by GPS certainly has advantages.