On the correct definition and use of normal heights in geodesy

Geodesy defines a physical height as the shortest distance of a point to a height reference surface in the Riemannian physical space. Two types of physical heights are most commonly used in science and applications, namely orthometric and normal heights. Both heights use the same reference surface (geoid) but differ in metrics associated with either the real (in case of orthometric heights) or model (for normal heights) gravity field. Orthometric heights correspond to the classical concept of physical heights introduced in the 19th century by Gauss, Stokes and Helmert. The main reason for introducing normal heights in the mid-20th century by Vignal and Molodensky, was related to a fact that the metric for orthometric heights (or corresponding gravity corrections to levelled height differences) could be estimated only approximately as they depend on the topographic mass density. Consequentially, normal heights have been introduced in many countries worldwide. However, their concept is sometimes misunderstood as the quasi-geoid is incorrectly referred to as their reference surface. Physical heights and corresponding height systems were discussed at the 10th Hotine-Marussi Symposium held in Milan, June 2022. A motion was proposed and discussed at the symposium that acknowledged reported problems of the quasi-geoid and its improper use as a height reference surface in geodesy. This presentation summarizes main arguments that were used to forge the motion.