# Geoid determination

Three-dimensional model of the “Potsdam Potato” (2017) with a 15,000-fold exaggerated representation of the geoid, German Research Center for Geosciences

With the term geoid determination , geodesy refers to processes that lead to the definition of the geoid . This task is closely related to the determination of details of the earth's gravity field . The methods for measuring the required physical parameters of the earth's gravitational field as well as the mathematical models for evaluating the recorded data are continuously being developed and are the subject of numerous research projects . The geoid determination is of fundamental importance for the national survey . The achievable accuracies have increased almost tenfold in the last five decades, so that today - depending on the effort - a few millimeters to centimeters would in principle be possible. Currently, the accuracy in the industrialized countries is 5–10 cm, while it will take about a decade to reach the centimeter geoid postulated around 1990 .

The geoid is that level surface of the earth's gravitational field that coincides with the mean sea ​​level of the world's oceans and continues in an abstract way at sea level among the continents. As an idealized shape of the earth's surface , it represents the “theoretical earth figure ” for the geosciences and serves as a reference surface for almost all height systems in use. Therefore, their precise determination is of the highest practical importance.

Earth's gravitational field: plumb line through P, perpendicular to it the geoid (V = Vo) and further equipotential surfaces (Vi)

## Global and regional geoid determination

Every concrete geoid determination usually refers to a mathematically clearly defined reference area, for which either the mean earth ellipsoid or the reference ellipsoid of the respective national survey is used. The distance to the selected ellipsoid, measured along the ellipsoid normal, is called the geoid height . The reference ellipsoids - of which around 200 are in use worldwide - were established in the industrialized countries around 100 years ago and fit as closely as possible to the geoid of the respective national territory. Depending on their location on the continents , they can vary in height by amounts between 10 and 100 meters from a globally averaged earth ellipsoid .

The axes of the global earth ellipsoid can now be calculated to within a few centimeters , to which satellite geodesy makes a significant contribution. Around 1960 they were known to about 30 meters, which, however, at an average earth radius of 6,370 km corresponds to a relative accuracy of five millionths (0.0005 percent or ½ cm / km). Local measurements within a radius of a few hectares to kilometers usually get by with this precision (e.g. 1 mm for technical buildings), but the demands on regional and global projects are increasing. With the Global Positioning System (GPS) or the Very Long Baseline Interferometry (VLBI) you can already reach the range from 10 −8 to 10 −9 , which in the last few decades has also required high-precision methods of geoid determination.

The shape of the geoid as a special level surface of the gravitational potential is shaped on the one hand by the global shape of the earth and the structure of the earth's interior , on the other hand by all the irregularities of the earth's crust . Accordingly, a distinction is made between "long-wave" structures - which can best be determined by evaluating satellite orbits - and the "short-wave" components that are shaped by the landscape and geology of the respective region. The latter - especially in the mountains  - can only be determined by local, terrestrial measurements.

The long-wave geoid undulations of the various continents are around 20 to 50 meters, the regional-local effects a few meters. However, you can - e.g. B. in the steep terrain of a mountainous country - change by a few decimeters per kilometer. In order to calculate such variable influences, in addition to a good surveying network, you also need an accurate terrain model and a digital model of the local rock density .

## Geoid determination as an interdisciplinary task

The most diverse areas of knowledge must therefore contribute to the modern methods of geoid determination: Mathematical geodesy and network adjustment , gravimetry and astronomical-geodetic measurement technology , radar technology and telemetry , the analysis of satellite orbits (see also satellite technology and orbital interference ), precise basics for the necessary reference systems , as well as data from geology and oceanography .

While the geoid on the oceans is represented by the mean sea level, which is thought to be at rest, it has to be imagined under the continents as its continuation - for example as the water level in an imaginary system of canals. It is only indirectly accessible for measurements here , so that the exact geoid determination has long been one of the most demanding tasks of higher geodesy . In the 1990s, the term centimeter geoid was coined for this challenge .

In contrast, the oceanic geoid is less precisely measurable, but it can be determined directly (e.g. by radar height measurement from satellites above sea level). The development of this satellite altimetry and its mathematical evaluation under the influence of the tides alone occupied a thousand scientists from various fields worldwide in the 1970s and 1980s.

## Physical, geometric and historical aspects

Although the geoid is a physically defined reference surface that results from the (variable) gravitational force and the rotation of the earth , it can be determined most precisely geometrically: namely by measuring the perpendicular direction to which it is everywhere. The classic geoid determination, the theory of which Friedrich Robert Helmert developed 120 years ago, looks for the surface that is perpendicularly penetrated by all measured perpendicular directions (see Potsdam gravity system ). Since the so-called vertical deviations are measured in the coordinate system of the stars, this oldest method of geoid determination is also called "astro-geodetic" and a regional geoid calculated with it is called " astrogeoid ".

In the 1940s, geoid determination was supplemented by physical methods, in particular by the development of accurate gravimeters for relative gravity measurement and the Eötvös rotary balance , which was also used for underground exploration of oil fields. At the same time, northern European geodesists developed the theory of isostasy (swimming equilibrium of mountains and continents), which made global calculations possible for the first time.

The combination of gravimetric and astro-geoid made it possible to calculate the global geoid to 5–10 meters in the middle of the 20th century, but on the mainland - depending on the measurement effort - to 20 to 100 centimeters.

Since the irregularities of the gravitational field and geoid are also reflected in the movement of earth satellites, a number of satellite geodesy methods and associated mathematical procedures for the so-called field continuation downwards (from the level of the satellite orbit down to the sea level) have been developed since the 1960s . They also required the establishment of new interdisciplinary research fields - for example collocation (joint treatment of geometrical-physical effects) or physical geodesy .

Summary: Because the geoid is the result of the physical mass distribution in the earth's interior as well as the topography (of the terrain ) and at the same time represents the (geometric) reference surface of our height systems, a wide variety of methods can be used to determine it. They fall into the following four groups of methods:

## Method groups for geoid determination

Vertical deviation: difference between the true vertical direction and a theoretical earth ellipsoid . It corresponds to the inclination between geoid and ellipsoid and distorts terrestrial surveying networks

### Measurements of gravity

1. terrestrial measurements with gravimeters
2. Gravity measurements in the aircraft

### Measurement of deviations from the perpendicular

1. Measurement of astro-geodetic vertical deviations
2. terrestrial - tacheometry or trigonometric altitude measurement
3. from astronomical azimuths in surveying networks (only relevant until around 1970)
Principle of satellite altimetry

### Measurements using earth satellites

3. with satellite altimetry (radar echoes over the sea , in future also over larger ice surfaces )

### Combination of some of the above procedures or data

1. astro-gravimetric geoid determination (TU Munich & Vienna, e.g. Prof. Hein, W. Daxinger)
2. Collocation of geometric and physical gravity field data
3. Interference potential by means of spherical function development plus terrestrial gravimetry

## literature

• Karl Ledersteger : Astronomical and physical geodesy (earth measurement) . JEK Volume V, Chapter 4 (vertical deviation and geoid determination), 7 (spherical functions) and 12/13 (Geoidundulations, Molodenskij, Weltsystem), JB Metzler-Verlag, Stuttgart 1968.
• Fernando Sansò , Michael Sideris (Eds.): Geoid Determination: Theory and Methods . (Advance info) Springer-Verlag, November 2007, ISBN 0-387-46386-0 , 300 pages.
• Gottfried Gerstbach : Regional geoid determination . Geoscientific Mitt. Volume 11, TU Vienna 1975.
• Siegfried Heitz : Geoid determination by interpolation according to smallest squares on the basis of measured and interpolated plumb line deviations . DGK series C, issue 124, Munich / Frankfurt 1968, 39 pages.
• Albrecht Preusser : A three-dimensional calculation model for geodetic point and geoid determination . DGK series C, issue 238, Munich 1977, 102 pages.