# Geoid

Earth's gravity field: Plumb line through surface point P, equipotential surfaces V i and the geoid (potential V = V o ) as a continuation of the mean sea level.

The geoid is an important reference surface in the earth's gravitational field . It is used to define heights as well as to measure and describe the earth figure . To a good approximation, the geoid is represented by the mean sea ​​level of the world's oceans and is thus directly visible in its shape outside the land masses .

The surfaces of the geoid are defined as the areas of equal gravitational potential . This makes the geoid surface at sea level the most informative, but all other surfaces are equivalent. The natural plumb line and the geoid surfaces are therefore perpendicular to each other at every point . Therefore, the geoid can be determined by measuring the acceleration due to gravity . Any two points on the geoid have the same gravitational potential and therefore the same dynamic height .

In contrast to the gravitational potential, the gravitational acceleration g is not constant on the geoid. Due to the increasing centrifugal acceleration from the pole to the equator, it sinks from 9.83 to 9.78 m / s². In addition, it varies locally due to the inhomogeneous mass distribution of the earth.

The geoid is a physical model of the earth figure, which was described by Carl Friedrich Gauß in 1828 - in contrast to the geometric model of the earth ellipsoid . The name geoid goes back to Johann Benedict Listing , who described it in 1871 as an area of ​​equal gravitational potential: The geoid is the equipotential area of ​​the earth's gravitational field at mean sea level, i.e. all points that have the same geopotential, composed of the gravitational potential and the Centrifugal potential at the relevant location. (see also: equipotential surface )

## Earth figure and geoid

The sea level is - apart from currents and tides - a so-called level surface , on which the gravitational potential is constant because it is everywhere perpendicular to the perpendicular. It is true that there are an infinite number of such equipotential surfaces that run like onion skins around the center of the earth . The special feature of the sea level, however, is that it can be observed across the globe through level observation and is therefore suitable as a global reference surface for height measurements and gravity measurements. For this purpose, some European countries set up and measured levels at various coastal locations around 200 years ago, for example the Amsterdam level or the level stations in Trieste , Genoa , Marseille and St. Petersburg . Their connection over land, made possible by height networks, would have been suitable for determining the continental geoid, but for political reasons this only happened with the European networks of the 20th century.

The regional determination of the geoid surface was carried out initially by astrogeodetic determining the perpendicular direction to individual survey points and from the 1930s through profile- or grid-like scale gravity measurements with gravimeters . Astrogeoids and gravimetric geoid determination have been noticeably improved by the Land Surveying Offices since around 1970 through strong compression of the vertical deviation or gravity networks , while global accuracy has been increased through years of satellite altimetry of the sea surface.

Today the automated methods of satellite geodesy dominate the determination of the earth's gravity field. They show the geoid as an irregular surface with many bumps and dents, but they only make up about 0.001 percent of the earth's radius . These wave-like geoid shapes are caused by anomalies in the gravity of the mountains and uneven mass distribution in the interior of the earth.

Due to its irregular shape, the geoid is very difficult to describe mathematically, whereas practical land surveying , cartography and GPS positioning require a more simply defined figure of the earth. Such reference surfaces for calculations and map images are mostly ellipsoids of revolution , which approximate the geoid with an accuracy of about 50 m . However, these strictly mathematical areas cannot be determined directly by measuring physical quantities .

Therefore, for practical use, the deviation between the physical earth figure (geoid) and its mathematical counterpart suitable for calculations (rotation ellipsoid) must be determined by systematic measurements . The deviations of the geoid from a reference ellipsoid (e.g. WGS84 , GRS 80 , Internationales Ellipsoid 1924 ) are referred to as geoid undulation or geoid height and can be up to 100 m and vary by about ± 30 m over 1000 km:

Geoidundulation , with ellipsoidal (geometric) height and orthometric (physical) height${\ displaystyle N = hH}$${\ displaystyle h}$${\ displaystyle H}$

## Geoid approximations with spherical functions

Pear shape as an approximation of the earth figure compared to the elliptical cross section (black line).
Illustration of the variation in gravity along the equator, based on a circular reference surface (black).

In the zeroth approximation , the geoid, neglecting the potential of the centrifugal force U z, is an equipotential surface in the gravitational field of a mass point: U ( r ) = G · M / r + U z ( G : gravitational constant , M : mass of the earth, r : distance from the center the earth). This simplification provides useful results for many calculations in celestial mechanics and space travel . The geoid is a sphere with a parameter R  ≈ 6373 km for the radius.

Deviations from the spherical shape can be described by Legendre polynomials P n (cos ( θ )) ( θ : latitude angle , R : mean earth radius, J n : expansion coefficients ):

${\ displaystyle U (r, \ theta) = {\ frac {GM} {r}} \ sum _ {n = 0.1, \ dots} \ left (\ left ({\ frac {R} {r}} \ right) ^ {n} J_ {n} \ cdot P_ {n} \ left (\ cos (\ theta) \ right) \ right) + U _ {\ mathrm {z}}}$

with the coefficients:

J 0 = 1; Ball approximation
J 1 = 0; no dipole moment, northern and southern hemisphere equally heavy
J 2 = 1082.6 x 10 -6 ; Approximate figure of the earth as an ellipsoid of revolution with equatorial semi-axes of equal size a  =  b  ≈ 6378 km and c  ≈ 6357 km as the polar axis. J 2 takes into account the so-called second order mass function , which comes from the flattening of the earth
J 3 = 2.51 x 10 -6 ; Put a pear-like structure on the ellipsoid (see drawing)
J 4 = 1.60 · 10 −6

The mass functions J 3 and J 4 cause geometric deviations from the mean earth ellipsoid that are less than 20 m. The high elevation in the drawing to the right illustrates why the earth is sometimes described as "pear-shaped".

An improved approximation introduces further spherical function coefficients, which take into account some of the dependencies of the geoid on the geographic longitude . The schematic drawing on the right makes it clear that there are deviations in gravity in the degree of longitude, which correspond to a height difference of 170 m. They are the reason why there are only two stable and two unstable orbit positions for geostationary satellites .

## Geoid determination

Measured deviations of the earth's gravitational field from the ellipsoid of revolution.
Three-dimensional model of the “Potsdam Potato” (2017) with a 15,000-fold amplified representation of the height deviation, German Research Center for Geosciences

The most precise determination of the entire geoid so far was carried out by the GRACE project . It consists of two satellites that orbit the earth at about 200 km distance at the same height. The distance between the two satellites is constantly measured with high accuracy. The shape of the geoid can then be deduced from the change in this distance.

The geoid determination can also be carried out with methods of astrogeodesy or gravimetrically; both provide the detailed forms of the geoid more precisely than the satellites, but are more complex. The determination of the astrogeoid (measurement of the vertical deviation ) was tested 100 years ago and is still the most accurate method today, but requires a survey network and clear nights for star observation. The ideal astrogeodesy instrument for this is the zenith camera : with its help, the perpendicular direction at a measuring point can be determined with high precision and partially automatically using CCD images of the zenithal star field . These plumb lines relate to the gravitational field and thus to the geoid. In order to determine the inclination of the geoid relative to the reference ellipsoid from deviations from the perpendicular , it is necessary to know the ellipsoidal coordinates of the measuring point. These can be determined from the national survey or with GNSS navigation satellites.

In gravimetry , the geoid is determined by measuring the acceleration due to gravity in a grid . However, the method is too complex for global geoid determination through a sufficiently dense distribution of the measuring points. For geoid interpolation between the measuring points, a digital terrain model is advantageous in the mountains - just as with the astrogeoid .

In June 2011 the German Research Center for Geosciences (GFZ) in Potsdam published the heavy model " EIGEN-6C ", which has become known as the Potsdam Potato . This global model was created from the combined data of various satellite measurements from LAGEOS , GRACE , GOCE and other measurement methods and has a spatial resolution of approx. 12 km.

## Causes of Geoid Undulations

Density anomalies in the earth's mantle due mantle and connected to them topography are variations of the cause of the major part of the observed geoid undulations.

The reasons for the long wavy geoid fluctuations (geoid undulations) are found in large-scale density variations in the earth's mantle and, to a lesser extent, in the earth's crust . An abnormally higher rock density creates an additional gravitational acceleration and thus bulges the geoid, lower densities lead to "dents" in the geoid. But the topography itself provides a laterally variable masses represents variation (→  uplift (geology) ) and results in undulations. The cause of density variations in the earth's mantle lies in mantle convection: hot mantle regions are less dense and rise (→  plume (geology) ); cold, dense regions sink.

One would now expect “dents” in the geoid from ascending convection currents, and “bumps” from descending convection currents (e.g. over subduction zones ), which by and large actually agrees with the observations for the West Pacific . However, things are made more complicated by the fact that rising convection currents can also raise the surface of the earth itself (e.g. Iceland , Hawaii ). The topography created in this way is called " dynamic topography". This weakens the actual negative geoid undulation and sometimes even reverses it into the positive range (which Iceland seems to be an example of). - Furthermore, the effect of the dynamic topography also depends on the viscosity of the earth's mantle and is difficult to quantify .

Today, findings from seismology in particular are used to estimate densities in the mantle and to calculate the geoid and dynamic topography. In this way, conclusions can be drawn about the jacket viscosity from the comparison with the observed geoid.

## Modern geoid solutions

Until around 1970, exact geoid determinations could almost exclusively be carried out on the mainland , which is why they are sometimes called regional geoid :

1. As an astrogeoid based on vertical deviations , obtained from a combination of astronomical and geodetic methods,
2. on the other hand as a gravimetric geoid by means of grid-shaped gravity measurements , as they are also required for geodetic precision leveling and in geophysics
3. or (since the 1970s) occasionally as a combined “astro-gravimetric geoid”.

With method (1), the distances between the measuring points were between about 10 km and 50 km, depending on the desired accuracy (5 cm to 50 cm), with (2, 3) about 3 to 15 km. The so-called centimeter geoid has been aimed for since around 1995 and has already achieved an accuracy of 2 to 3 cm in some countries in Central Europe.

With the increasing success of satellite geodesy , models of geopotential (gravitational field in the outer space of the earth) also contributed to geoid determination. From the orbital disturbances caused by the geoid and the interior of the earth , high-grade potential developments with spherical surface functions were calculated, which initially had a resolution of around 20 degrees of latitude and longitude (around 1000 km × 1000 km), but today down to 0.5 ° (around 50 km).

The first spherical function developments had global accuracies of about 10 m, which today have improved to well below 1 m (that's about 0.00001% of the Earth's radius ). In contrast to the methods mentioned above, they cannot resolve any details, but they can support a regional geoid to the outside and enable the merger to form continental solutions. Satellite-to-satellite tracking (STS) is the newest method today .

## literature

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• Erwin Groten: Geodesy and the Earth's Gravity Field. Volume I: Principles and Conventional Methods. Bonn 1979.
• Karl Ledersteger : Astronomical and physical geodesy (= handbook of surveying. Volume 5). 10th edition, Metzler, Stuttgart 1969.
• Gottfried Gerstbach : How to get an European centimeter geoid (“astro-geological geoid”). In: Physics and Chemistry of the Earth. Volume 21/4. Elsevier, 1996, pp. 343-346.
• Heiner Denker, Jürgen Müller et al .: A new Combined Height Reference Surface for Germany ( GCG05 ). EUREF Conference, Riga 2006, ( poster ; PDF; 414 kB).
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