Residual Terrain Modelling: The Harmonic Correction for Geoid Heights

The harmonic correction (HC) is one of the key quantities when using residual terrain modelling (RTM) for high-frequency gravity field modelling. In the RTM technique, high-frequency topographic gravitational signals are obtained through removing gravitational effects of a long-wavelength reference surface, e.g., MERIT2160. There might be points located below the reference surface. In such cases, the RTM gravity field is calculated in the non-harmonic condition, HC is therefore required. Over past decades, though various methods have been proposed to handle the HC issue for the RTM technique, most of them were focused on the HC for RTM gravity anomaly rather than for other gravity functionals, such as RTM geoid height. In practice, the HC for RTM geoid height was generally assumed to be negligible, but a detailed quantification was missing for present-day RTM computations. This might cause large errors in the regional geoid determination over rugged areas. In this study, we derive HC expressions for the RTM geoid height in the framework of the classical condensation method. The HC terms are derived under four different assumptions separately: residual masses approximated by an unlimited Bouguer plate, residual masses approximated by a limited Bouguer plate which overcomes the mass inconsistency effect, residual masses approximated by a Bouguer shell which overcomes the effect of planar approximation, and residual masses approximated by a limited Bouguer shell which overcomes the errors induced by both planar approximation and mass-inconsistency. The errors due to various approximations in HC terms are investigated through comparison among various terms. Besides, HC terms are computed using an expansion up to degree and order 2159. Our results show that HC for RTM geoid height is less 1 mm and could be ignored over \(\sim 99\)% of continental areas, but be of great significance for regional geoid determination over mountain areas, e.g., more than 10 cm effect over very rugged areas. The validation through comparison with terrestrial measurements and a baseline solution of the RTM technique proves that the HC terms provided in this study can improve the accuracy of RTM geoid heights and are expected to be useful for applications of the RTM technique in regional and global gravity field modelling.