The Terrain Correction in a Moving Tangent Space

Conventionally the terrain/topographic reduction is based on the Bouguer Plate, which is flat and extends in the local tangent plane/horizontal plane to infinity. Here we aim at an error estimate of such a planar approximation of the Newton integral of the type of a disturbing potential and gravitational disturbance as linearized forms of the gravitational potential and the modulus of gravitational field intensity. To effect this quality control of the conventional terrain reduction, we first transform the spherical Newton functional from an equatorial frame of reference to an oblique meta-equatorial frame of reference with the evaluation point as a meta-North pole, and then by means of an oblique equiareal map projection of the azimuthal type to a tangent plane which moves at the evaluation point. The first term of these transformed Newton functionals is the planar approximation. The difference between the exact Newton kernels and their planar approximation are plotted and tabulated in Tables 1-3. Three configurations are studied in detail: for points at radius r = 10 km around the evaluation point the systematic error varies from 0.26% for a spherical height difference of the order of H – H* = 5 km, more than 0.80% for a spherical height difference of the order of H – H* = 1 km, and more than 1.60% for a spherical height difference of H – H* = 500 m. In contrast, the systematic error for spherical height difference H – H* = 1 km at a distance of r = 1000 km from the evaluation point increases to 44%. Indeed, the newly derived exact Newton kernels which are of the convolution type and are represented in the tangent space moving with the evaluation point can be preferably used with little extra computational effort.