Application of M¨obius coordinate transformation in evaluating Newton’s integral

Abstract

We propose a numerical scheme which efficiently combines various existing
methods of solving the Newton’s volume integral. It utilises the analytical solution of
Newton’s integral for tesseroid in computing the near-zone contribution to gravitational
field quantities (potential and its first radial derivative). The far-zone gravitational contribution
is computed using the expressions derived based on applying Molodensky’s truncation
coefficients to a spectral representation of Newton’s integral. The weak singularity
of Newton’s integral is treated analytically using formulas for the gravitational contribution
of the cylindrical mass volume centered with respect to the observation point. All
three solutions are defined and evaluated in the system of polar spherical coordinates. A
conversion of the geographical to polar spherical coordinates of input data sets (digital
terrain and density models) is based on the M¨obius transformation with an enhanced
integration grid resolution at vicinity of the observation point.