Cap integration in spectral gravity forward modelling up to the full gravity tensor

Abstract

Cap-modified spectral gravity forward modelling is extended in this paper to the full gravity vector and tensor expressed in the local north-oriented reference frame. This is achieved by introducing three new groups of altitude-dependent Molodensky’s truncation coefficients. These are given by closed form and infinite spectral relations that are generalized for (i) an arbitrary harmonic degree, (ii) an arbitrary topography power, (iii) an arbitrary radial derivative, (iv) any radius larger than the radius of the reference sphere, and (v) for both near- and far-zone gravity effects. Thanks to the generalization for an arbitrary radial derivative, the cap-modified technique can efficiently be combined with the gradient approach for harmonic synthesis on irregular surfaces. In a numerical study, we exemplarily apply the new technique by forward modelling Earth’s degree-2159 topography up to degree 21,590, employing 30 topography powers. The experiment shows that near- and far-zone gravity effects can be synthesized on the topography with an accuracy (RMS) of 0.005–0.03 m $$^2$$ s $$^{-2}$$ (potential), 0.8–20 $$\upmu \mathrm {Gal}$$ (gravity vector), and 0.1 mE–1 E (gravity tensor). The numerical experiment also shows that the divergence effect of spherical harmonics comes into play around degree 10,795 when evaluating the series on the Earth’s surface. The difficult-to-compute truncation coefficients that are employed in the study are made freely available at http://edisk.cvt.stuba.sk/~xbuchab/ and are accompanied by MATLAB-based routines to evaluate them. Enclosed is also a MATLAB-based package to perform ultra-high-degree surface spherical harmonic analysis, a step of central importance in spectral gravity forward modelling techniques.