On the effect of planar approximation in the Geodetic Boundary Value Problem

Abstract

In the framework of the analytical treatment of the Geodetic Boundary Value Problem (GBVP) various approximations are necessary in order to end up with a solution in a “closed” form. Besides linearization and isotropic (“spherical”) approximation the socalled planar approximation has been introduced in the case of a non-spherical, topographic boundary surface. In this context the term “planar approximation” refers to neglecting terms proportional to the quotient of elevation and mean Earth radius in the respective kernel functions, not to confuse with approximating the boundary surface by a plane. In this paper the effect of planar approximation on the boundary condition of the fixed GBVP and on the disturbing potential is estimated. A global study is presented on the basis of synthesized gravity disturbances computed from the global geopotential model EGM2008 and the topography model DTM2006.0. A regional densification is carried out for the area of the Andes; therefore the gravity disturbances are refined by RTM-effects on the basis of the SRTM3 topography model. This analysis which is valid for the scalar free and the fixed versions of the GBVP simultaneously, is based on high-resolution digital elevation models (DEM) and shows that the numerical estimates of the effects of planar approximation depend on the resolution of the DEMs. The numerical investigations confirm the assumed relative error of the order 0.1%. This assumption implies a maximum effect of approximately 0.5 mGal for the boundary values and 10 cm for the quasigeoidal heights.