On the geoid and orthometric height vs. quasigeoid and normal height

Lars E Sjöberg

doi:10.1515/jogs-2018-0011

Abstract

Abstract The geoid, but not the quasigeoid, is an equipotential surface in the Earth’s gravity field that can serve both as a geodetic datum and a reference surface in geophysics. It is also a natural zero-level surface, as it agrees with the undisturbed mean sea level. Orthometric heights are physical heights above the geoid,while normal heights are geometric heights (of the telluroid) above the reference ellipsoid. Normal heights and the quasigeoid can be determined without any information on the Earth’s topographic density distribution, which is not the case for orthometric heights and geoid. We show from various derivations that the difference between the geoid and the quasigeoid heights, being of the order of 5 m, can be expressed by the simple Bouguer gravity anomaly as the only term that includes the topographic density distribution. This implies that recent formulas, including the refined Bouguer anomaly and a difference between topographic gravity potentials, do not necessarily improve the result. Intuitively one may assume that the quasigeoid, closely related with the Earth’s surface, is rougher than the geoid. For numerical studies the topography is usually divided into blocks of mean elevations, excluding the problem with a non-star shaped Earth. In this case the smoothness of both types of geoid models are affected by the slope of the terrain,which shows that even at high resolutions with ultra-small blocks the geoid model is likely as rough as the quasigeoid model. In case of the real Earth there are areas where the quasigeoid, but not the geoid, is ambiguous, and this problem increases with the numerical resolution of the requested solution. These ambiguities affect also normal and orthometric heights. However, this problem can be solved by using the mean quasigeoid model defined by using average topographic heights at any requested resolution. An exact solution of the ambiguity for the normal height/quasigeoid can be provided by GNSS-levelling.

Highlights

The geoid is an important equipotential surface and vertical reference surface in geodesy and geophysics
We show from various derivations that the difference between the geoid and the quasigeoid heights, being of the order of 5 m, can be expressed by the simple Bouguer gravity anomaly as the only term that includes the topographic density distribution
For numerical studies the topography is usually divided into blocks of mean elevations, excluding the problem with a non-star shaped Earth. In this case the smoothness of both types of geoid models are affected by the slope of the terrain, which shows that even at high resolutions with ultra-small blocks the geoid model is likely as rough as the quasigeoid model

Summary

Introduction

The geoid is an important equipotential surface and vertical reference surface in geodesy and geophysics. As the quasigeoid is closely related with the Earth’s geometric surface, Vanicek et al (2012) raised the question whether it can be practically determined according to Molodensky’s proposed method by successive approximations. One may still believe that it is much more irregular than the geoid This problem will be analysed by comparing formulas for geoid and quasigeoid determination. The density is assumed to be constant, and the geoid and quasigeoid are determined only from surface gravity data by Stokes-types formulas. The LSMSA technique is based on analytical continuation of the surface gravity anomaly to sea-level and surface level in geoid and quasigeoid determinations, respectively The combined topographic effect is the same as the negative of the topographic bias (Sjöberg 2007 and 2009a, b; Sjöberg and Bagherbandi 2017, Sects. 5.2.3-5.2.4): dN

Determination of the geoid

Findings

GNSS-levelling approach