Error propagation for three common height-system corrections to differential levelling

This paper investigates the propagation of input data errors through the application of Helmert orthometric, normal and normal-orthometric height corrections to differential levelling observations, these being the three principal height systems in practical use around the world. Height corrections are required to remove the systematic error resulting from the geometric non-parallelism of the Earth's equipotential surfaces, but different height systems propagate errors differently. These systematic errors are thus present within levelling networks and subsequently in local vertical datums. Here, we show that the Helmert orthometric correction is sensitive to errors in the mean value of gravity along the plumbline, particularly for heights above 1000 m. The normal correction is much less sensitive due to the use of normal gravity along the normal plumbline. The normalorthometric correction of Rapp (1961) is largely insensitive to such errors, but it does not properly correct for the non-parallelism of the Earth's equipotential surfaces. Information showing the circumstances under which survey practitioners should apply height corrections to levelling lines is provided, demonstrating that normal-orthometric corrections only need be applied to class LC levelling lines that are to be used for large levelling networks extending in the north-south direction, particularly at high elevations.