Ellipsoidal approximation of the topographical effects in the Earth's gravity field modelling



The topographical effect is one component of the Earth's gravity field that needs to be reliably evaluated in the gravity field modeling. The topographical effect can be numerically evaluated from the knowledge of a Digital Terrain Models (DTM). After the Satellite positioning system, e.g., GPS, the computation points as well as DTMs present/convert in Gauss ellipsoidal (geodetic) coordinates system, λ, φ and h called ellipsoidal longitude, longitude and height, respectively.
 So far, the planar and spherical models of the topography are frequently used for computation of the effect of topographical masses in geodesy and geophysics. In practice, the planar model is widely used in the evaluation of the classical terrain correction. Vanicek et al. (2001) indicated that the planar model of topography (in form of infinite Bouguer plate) cannot be applied for the solution of the geodetic boundary value problem. Also, spherical approximation of the topography may be insufficient for precise determination of the 1cm-geoid. Moreover, the interested points on and above the Earth’s surface as well as the DTMs are presented in geodetic coordinate system. Therefore the Newton's integral and related formulas should be evaluated in terms of the geodetic coordinates system. In this study, a new exact ellipsoidal formula for potential of topography and its vertical gradient, as well as for second Helmert condensation topography effects are derived.
The Newton's integral for computation of the gravitational potential and its vertical gradient has a weak singularity when the computation point is close to the integration point. According to Martinec (1998), the singularity is removed from the numerical integration using the Cauchy algorithm by adding and subtracting the Bouguer terms (the singularity contribution). In ellipsoidal approximation, the Bouguer terms are computed from an ellipsoidal shell. The ellipsoidal shell is sufficiently approximated by a shell bounded by two concentric, similar ellipsoids that so called homoeoid. The thickness of homoeoid is equal to ellipsoidal height of topography at the interest point. The roughness terms, due to deficiency of the ellipsoidal Bouguer shell can be evaluated by direct numerical integration.    
The results of two spherical and ellipsoidal models are numerically investigated in Iran (the highest peak exceeds 5000 m). The selected test area extends from 24° to 40° northern latitudes and from 44° to 60° eastern longitudes. Near zone of topographical integrals extends to 4° and the far zone from 4° to 180°. Near distant is divided into three zones. 1- Innermost zone to 15 minute, 2- middle zone to 1°, and outer zone from 1° to 4°. The contribution of Innermost, middle and outer zones is computed by 3", 30" and 5' DEMs. Far zone effect is computed by integration over a 30' DTM. The numerical results indicate that the magnitudes of ellipsoidal corrections (difference between ellipsoidal and spherical solutions) are small.  The main bulk of this correction is long wavelength and is due to Bouguer and distance zone contributions. Therefore the ellipsoidal correction can be sufficiently used for regional and global applications such as regional Earth's gravity field approximation. Since for the compilation of 1cm geoid, the gravity with a precision better than 10 µGal is needed (Martinec, 1998), the ellipsoidal approximation of topography must be used in precise geoid computation particularly in rugged mountainous area.