Exact closed-form expressions for the complete RTM correction

We present exact, closed-form expressions for the complete RTM correction and the harmonic correction to disturbing potential, gravity disturbance, gravity anomaly, and height anomaly. They need to be applied in quasi-geoid modelling whenever data points are buried inside the masses after residual terrain model (RTM) reduction and analytically downward continued functionals of the disturbing potential at the original locations of the data points are required. Compared to recent work of the authors published in this journal, no Taylor series enter the expressions and numerical instabilities of the harmonic downward continuation from the RTM surface to the Earth's surface are avoided as are inaccuracies in the free-air upward continuation from the Earth's surface to the RTM surface caused by a lack of precise information about higher-order derivatives of the disturbing potential. The new expressions can easily be implemented in any existing RTM software package and do not require additional computational resources. For two test areas located in western Norway and the Auvergne in France, we compute the complete RTM correction and the harmonic correction to the afore-mentioned functionals of the disturbing potential. Overall, all harmonic corrections are non-negative with maximum values of 1.54 m(2)/s(2), 263.0 mu Gal, 263.9 mu Gal, and 15.7 m (Norway) and 1.55 m(2)/s(2), 263.3 mu Gal, 263.3 mu Gal, and 15.8 cm (Auvergne) for disturbing potential, gravity disturbance, gravity anomaly, and height anomaly, respectively. The medians are 0.02 m(2)/s(2), 33.6 mu Gal, 33.7 mu Gal, and 0.3 cm (Norway) and 0.01 m(2)/s(2), 19.2 mu Gal, 19.2 mu Gal, and 0.1 cm (Auvergne). We also show that the nth Taylor polynomials used in the recent work of the authors published in this journal may have large remainders depending on the topography in the vicinity of the evaluation point no matter how n is chosen. Finally, we show that the commonly used expression for the harmonic correction to gravity anomaly introduced in 1981 is almost exact, though it was derived along a completely different line of reasoning. The errors do not exceed 49 mu Gal in both test areas. Moreover, the errors have a negligible impact on the computed height anomalies in one-centimetre quasi-geoid modelling, as the mean error does not exceed a few mu Gal in both test areas.