Geoid

A geoid is an important reference surface in the gravity field of the Earth. It is used to define heights as well as for the measurement and description of the earth's shape. In good approximation, the geoid is represented by the mean sea level of the oceans and is therefore directly visible in his form beyond the land masses.

The gravity potential is the same at every location of the geoid surface. The natural perpendicular and the geoid are perpendicular to each other at any point. Therefore, the geoid can be determined by measuring the gravitational acceleration. Any two points on the geoid have the same gravity potential and therefore the same dynamic height.

In contrast to the gravity potential is the gravitational acceleration g on the geoid is not constant but decreases due to the centrifugal acceleration from the pole to the equator rising from the pole to the equator of 9.83 to 9.78 m/s2.

The geoid is a physical model of the Earth's figure, which was developed in 1828 by Carl Friedrich Gauss - in contrast to the geometric model of the earth ellipsoid. The term " geoid " goes back to Johann Benedict Listing, who described it in 1871 as a surface of equal gravity potential.

Earth's figure and geoid

The sea level is - apart from currents and tides - a so-called level surface on which the gravity potential is constant, because it everywhere perpendicular to the plumb line. Although there are infinitely many such equipotential surfaces, run like onion shells around the center of the earth. However, the sea level has the peculiarity that it is erdumspannend observed by observation level and is intended as a global reference surface for height measurements and gravity measurements. To this end, some European countries have already established level about 200 years ago in various coastal towns and measured, such as the Amsterdam level or the level stations in Trieste, Genoa, Marseilles and St. Petersburg. Your potential height by networks connect overland would have been appropriate for the determination of the continental geoid, but this was for political reasons only with the European networks of the 20th century.

The regional determination of the geoid surface was initially astrogeodätische by determining the perpendicular direction to individual survey points and from the 1930s through profile - or raster-like scale gravity measurements with gravimeters. From the offices of the provincial surveying the Astrogeoide and the gravimetric geoid determination has been since about 1970 markedly improved by strong compression of Lotabweichungs or gravity networks, while the global accuracy was increased by years of satellite altimetry of the ocean surface.

Today, the automated method of satellite geodesy dominate the determination of the Earth's gravity field. They show the geoid is an irregular surface with many dents and dings, but account for only about 0.001 percent of the Earth's radius. This wave-like Geoidformen caused by gravity anomalies of the mountains and uneven distribution of mass within the earth.

Because of its irregular shape of the geoid is mathematically very difficult to describe, whereas the practical land surveying, cartography and GPS positioning need a simple defined figure of the earth. Such reference surfaces for calculations and map images are mostly rotational ellipsoids, which approximate the geoid to about 50 meters. However, this strict mathematical surfaces can not be determined directly by measuring physical quantities.

It must therefore be determined by systematic measurements for the practical handling of the deviation between the physical figure of the earth ( geoid ) and their mathematical, suitable for calculations Pendant ( spheroid ). The deviations of the geoid by a reference ellipsoid (eg WGS84, GRS 80 ellipsoid International 1924) are referred to as geoid undulation or geoid height and can make up to 100 m and vary by 1,000 km by about ± 30 meters:

Geoid approximations with spherical harmonics

In the zeroth approximation, the geoid is neglecting the potential of the centrifugal force Uz an equipotential surface in the gravity field of a mass point: U ( r) = G * M / r Uz (G: gravitational constant, M: mass of the Earth, r: distance from the center of the earth ). For many calculations in celestial mechanics and space, this simplification provides useful results. 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 Pn (cos ( θ )) describe ( θ: Wide angle, R: mean radius of the earth, Jn: expansion coefficients ):

With the coefficients:

The mass functions J3 and J4 cause geometric deviations from the mean Erdellipsoid that are less than 20 m. The strong increase in the drawing on the left illustrates why the earth is manchesmal described as " pear-shaped ".

An improved approximation results in a more spherical function coefficients that take into account some dependencies of the geoid from the longitude. The diagram on the right shows that gravity variations present in longitude, corresponding to a height difference of 170 m. They are the reason that there are only two stable and two unstable orbit position for geostationary satellites.

Geoid determination

The most precise determination of the total geoid was made by the GRACE project. It consists of two satellites orbiting at approximately 200 km distance in the same amount to earth. The distance between the two satellites is continuously measured with high accuracy. From the change of this distance then one concludes that the shape of the geoid.

The geoid determination can also be made by methods of Astro geodesy or gravimetrically; both provide the detailed shapes of the geoid accurate than the satellite, but are more expensive. The determination of the Astrogeoids (measurement of the deflection of the vertical ) was tested 100 years ago and is still the most accurate method, but requires a survey network and clear nights for stargazing. The optimal instrument for the Astro Geodesy is the Zenit Camera: with its help can be determined with high precision in a measurement point and partially automated by CCD images of the zenithal star field the perpendicular direction. This Lotrichtungen refer to the gravitational field and thus on the geoid. To determine the slope of the geoid from deflections of the vertical with respect to the reference ellipsoid, the knowledge of the ellipsoidal coordinates of the measuring point is required. These can be determined from the national survey or with GNSS navigation satellites.

In the gravimetric geoid is determined by grid-like measurement of the gravitational acceleration. For a global geoid determination by sufficiently dense distribution of the measurement points, the method is, however, too complex. To geoid interpolation between measurement points is in the mountains - as well as in Astrogeoid - a digital terrain model advantageous.

In June 2011, the GFZ published the gravity model " SELF -6C ". This global model was created from the combined data of different satellite measurements of LAGEOS, GRACE, GOCE and other measurement methods and has a spatial resolution of about 12 km.

Causes of geoid undulations

Density anomalies in the mantle due to mantle convection and their affiliates topography variations are the cause of the majority of the observed geoid undulations.

The causes of the long-wave Geoidschwankungen ( geoid undulations ) are in large-scale density variations in the mantle and to a lesser extent in the earth's crust. An abnormally higher rock density generates an additional gravitational acceleration and thus bulges from the geoid, lower densities lead to "bump " in the geoid. However, the topography itself is an laterally variable mass variation is (→ uplift (geology) ) and leads to undulations. The cause of density variations in the mantle is the mantle convection: Hot cladding regions are less dense and rise (→ Plume (geology) ); cold, dense regions from falling. So you would now expect over rising convection currents " dents " in the geoid, on subducting convection (eg via subduction zones ) " bumps ", which actually matches for the Western Pacific, on the whole with the observations. The thing is, however, more complicated that rising convection currents and the earth's surface can lift itself (eg: Iceland, Hawaii). The topography thus produced is referred to as " dynamic topography ". Result, the actual negative geoid undulation is weakened and in some cases even to positive territory vice versa (for which Iceland seems to be an example). - Furthermore it depends on the effect of dynamic topography but also on the viscosity of the mantle and is difficult to quantify.

Today it draws on insights especially from seismology to estimate densities in the shell and to calculate the geoid and the dynamic topography. From the comparison with the observed geoid conclusions can thus draw on the mantle viscosity.

Modern Geoidlösungen

Until about 1970, exact Geoidbestimmungen could be carried out almost exclusively on the mainland, which is why they are sometimes called Regional geoid:

In method ( 1) the distances between the measurement points were depending on the desired accuracy ( 5 cm to 50 cm) between 10 km and 50 km, at (2, 3) about 3 to 15 km. Since about 1995 it seeks to achieve the so-called centimeter geoid and today has already reached 2-3 cm accuracy in some countries of Central Europe.

With the increasing success of satellite geodesy also contributed

• Models of the geopotential ( gravitational field in the exterior of the earth ) at the geoid determination. From the caused of geoid and Erdinnerem perturbations high grade potential developments were calculated using spherical harmonics, which initially about 20 latitude and longitude resolution had ( about 1000 × 1000 km ), while today down to 0.5 ° ( about 50 km).

The first spherical harmonic developments had global accuracies of about 10 meters, which has now improved to well below 1 meter ( or approximately 0.00001 % of the Earth's radius ). In contrast to above-mentioned methods, although they can not resolve details, but probably support a regional geoid to the outside and allow the merger to continental solutions. The latest method is used today

• The Satellite -to -Satellite Tracking ( STS)