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 μ Gal, 263.9 μ Gal, and 15.7 m (Norway) and 1.55 m ²/ s ², 263.3 μ Gal, 263.3 μ 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 μ Gal, 33.7 μ Gal, and 0.3 cm (Norway) and 0.01 m ²/ s ², 19.2 μ Gal, 19.2 μ 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 μ 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 μ Gal in both test areas.

In this paper, we derive improved expressions for the harmonic correction to gravity and, for the first time, expressions for the harmonic correction to potential and height anomaly. They need to be applied at stations buried inside the masses to transform internal values into harmonically downward continued values, which are then input to local quasi-geoid modelling using least-squares collocation or least-squares techniques in combination with the remove-compute-restore approach. Harmonic corrections to potential and height anomaly were assumed to be negligible so far resulting in yet unknown quasi-geoid model errors. The improved expressions for the harmonic correction to gravity, and the new expressions for the harmonic correction to potential and height anomaly are used to quantify the approximation errors of the commonly used harmonic correction to gravity and to quantify the magnitude of the harmonic correction to potential and height anomaly. This is done for two test areas with different topographic regimes. One comprises parts of Norway and the North Atlantic where the presence of deep, long, and narrow fjords suggest extreme values for the harmonic correction to potential and height anomaly and corresponding large errors of the commonly used approximation of the harmonic correction to gravity. The other one is located in the Auvergne test area with a moderate topography comprising both flat and hilly areas and therefore may be representative for many areas around the world. For both test areas, two RTM surfaces with different smoothness are computed simulating the use of a medium-resolution and an ultra-high-resolution reference gravity field, respectively. We show that the errors of the commonly used harmonic correction to gravity may be as large as the harmonic correction itself and attain peak values in areas of strong topographic variations of about 100 mGal. Moreover, we show that this correction may introduce long-wavelength biases in the computed quasi-geoid model. Furthermore, we show that the harmonic correction to height anomaly can attain values on the order of a decimetre at some points. Overall, however, the harmonic correction to height anomaly needs to be applied only in areas of strong topographic variations. In flat or hilly areas, it is mostly smaller than one centimetre. Finally, we show that the harmonic corrections increase with increasing smoothness of the RTM surface, which suggests to use a RTM surface with a spatial resolution comparable to the finest scales which can be resolved by the data rather than depending on the resolution of the global geopotential model used to reduce the data.

We study the impact of an accurate computation and incorporation of coloured noise in radar altimeter data when computing a regional quasi-geoid model using least-squares techniques. Our test area comprises the Southern North Sea including the Netherlands, Belgium, and parts of France, Germany, and the UK. We perform the study by modelling the disturbing potential with spherical radial base functions. To that end, we use the traditional remove-compute-restore procedure with a recent GRACE/GOCE static gravity field model. Apart from radar altimeter data, we use terrestrial, airborne, and shipboard gravity data. Radar altimeter sea surface heights are corrected for the instantaneous dynamic topography and used in the form of along-track quasi-geoid height differences. Noise in these data are estimated using repeat-track and post-fit residual analysis techniques and then modelled as an auto regressive moving average process. Quasi-geoid models are computed with and without taking the modelled coloured noise into account. The difference between them is used as a measure of the impact of coloured noise in radar altimeter along-track quasi-geoid height differences on the estimated quasi-geoid model. The impact strongly depends on the availability of shipboard gravity data. If no such data are available, the impact may attain values exceeding 10 centimetres in particular areas. In case shipboard gravity data are used, the impact is reduced, though it still attains values of several centimetres. We use geometric quasi-geoid heights from GPS/levelling data at height markers as control data to analyse the quality of the quasi-geoid models. The quasi-geoid model computed using a model of the coloured noise in radar altimeter along-track quasi-geoid height differences shows in some areas a significant improvement over a model that assumes white noise in these data. However, the interpretation in other areas remains a challenge due to the limited quality of the control data.