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.

在本文中，我们推导了重力谐波校正的改进表达式，并首次导出了位势和高度异常的谐波校正表达式。它们需要应用在埋藏在群众内部的台站，以将内部值转换为谐波向下的连续值，然后使用最小二乘搭配或最小二乘技术结合删除-计算-恢复将其输入到局部准大地水准面建模方法。到目前为止，假设对潜在和高度异常的谐波校正可以忽略不计，从而导致未知的准大地水准面模型误差。重力谐波校正的改进表达式，采用新的位高异常调和改正表达式，量化了常用的重力调和改正的逼近误差，量化了位高异常调和改正的幅度。这是针对具有不同地形状况的两个测试区域完成的。其中之一包括挪威和北大西洋的部分地区，那里深、长和窄的峡湾的存在表明对潜在和高度异常的谐波校正具有极值，以及对重力的谐波校正的常用近似值的相应大误差。另一个位于奥弗涅测试区，地形适中，包括平坦和丘陵地区，因此可能代表世界上许多地区。对于两个测试区域，计算两个具有不同平滑度的 RTM 表面，分别模拟中分辨率和超高分辨率参考重力场的使用。我们表明，常用的谐波校正对重力的误差可能与谐波校正本身一样大，并且在大约 100 mGal 的强烈地形变化区域达到峰值。此外，我们表明这种校正可能会在计算的准大地水准面模型中引入长波长偏差。此外，我们表明，对高度异常的谐波校正可以在某些点达到分米量级的值。然而，总的来说，对高度异常的谐波校正只需要在地形变化很大的区域应用。在平坦或丘陵地区，它大多小于一厘米。最后，