Comparison of two different approaches for computing the gravitational effect of a tesseroid

Forward modelling in the space domain is a very important task in geodesy and other geosciences. From topographical or isostatic information in the form of digital terrain model (DTM) and density model, the effects of these parameters or their derivatives on the gravity potential can be evaluated for different applications. In most cases, height or height-layer models are in use, which are gridded with respect to spherical coordinates. This holds for global as well as regional or even local applications. The definition of the spherical gridlines leads immediately to the spherical volume element, that is, the tesseroid. Only in the specific case that the observation point is located on the symmetry axis of the spherical coordinate system does the Newton integral have a closed analytical solution. More specifically, the effect of a tesseroid can be determined by evaluating the analytical solution of a segment of a spherical zonal band. To apply this aspect in practice, the DTM must be transformed into the local spherical azimuthal system of the observation point (UNIPOL approach). In the general case, the Newton integral can be solved, for example, using a Taylor series expansion of the integral kernel and a subsequently applied term-wise integration (GIK approach). Within this contribution, the two fundamentally different tesseroid approaches, namely, the GIK and the UNIPOL approach are compared. This comparison is performed, in particular, with regard to the required computational time and the approximation error under different test scenarios. The numerical studies show that both approaches are equivalent in terms of accuracy for both the gravitational potential and gravity; however, the UNIPOL approach is more time consuming because, for each observation point, the whole DTM must be transformed. Small numerical differences exist between the compared approaches for special constellations regarding the source point and the observation point.