The terrain correction in a moving tangent space

Conventionally the terrain/topographic reduction is based on the Bouguer Pl ate, 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 approximati on" of the Newton integral of the type of a disturbing potential and gravit ational disturbance as linearized forms of the gravitational potential and the modulus of gravitational field intensity. To effect this quality v cont rol 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 po le, and then by means of an oblique equiareal map projection of the azimuth al type to a tangent plane which moves at the evaluation point. The first t erm of these transformed Newton functionals is the "planar approximation". The difference between the exact Newton kernels and their "planar approxima tion" are plotted and tabulated in Tables 1-3. Three configurations are stu died 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 kin, more than 0.80% for a spherical height differ ence of the order of H - H* = I kin, and more than 1.60% for a spherical he ight difference of H - H* = 500 ni. In contrast, the systematic error for s pherical height difference H - H* = I km at a distance of r = 1000 kin from the evaluation point increases to 44%. Indeed, the newly derived exact New ton kernels which are of the convolution type and are represented in the ta ngent space moving with the evaluation point can be preferably used with li ttle extra computational effort.