Astrogeoid

from Wikipedia, the free encyclopedia

As Astrogeoid one is geoid determination with astro-geodetic measurements referred. In its classic version, it is based on a profile-like or areal integration of measured perpendicular deviations . As the forerunners of the methodology, those in the 18th – 19th Century carried out exact degree measurements to determine the theoretical figure of the earth .

The methodology is also called "Astronomical leveling called" because it is an analogue to trigonometric leveling is where the astronomically measured plumb the role of the elevation angle takes over. The theory was developed 130 years ago (by the German geodesist Friedrich Robert Helmert ) and its basic features can be traced back to Carl Friedrich Gauß .
Their first satisfactory application (in the Harz Mountains and in Austria ) only succeeded at the beginning of the 20th century with the development of practicable measurement methods in astrogeodesy . To this day it is the most precise method for geoid determination on the mainland , which can reach a few mm to cm. Today it is increasingly combined with gravimetry in national surveying projects , while satellite geodesy is the preferred method in the oceans .

A geoid determination related to the astro-geoid is that with gravity gradients . In some countries (e.g. Hungary) attempts are being made to combine the too thin vertical deviation network with rotary balance measurements from oil exploration or with gradiometry (see web links).

Plumb deviation and geoid

The geoid is the level surface of the earth that coincides with the mean sea ​​level of the world's oceans. As an idealized shape of the earth's surface , it represents the “theoretical earth figure ” for geosciences and serves as a reference surface for almost all height measurements. Therefore, their precise determination is of the highest practical importance.

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

The perpendicular deviation is the deviation of the true perpendicular from the normal of a mathematical reference surface , for which a regional or global earth ellipsoid is usually used. The difference between the determined astronomical latitude  φ or length λ and the ellipsoid normal ( geodetic latitude  B and longitude  L ) is in the two components

ξ = φ - B
η = (λ - L) .cos φ

specified. These north-south or east-west components of the perpendicular deviation can be determined with an accuracy of 0.1 ″ to 0.3 ″, depending on the effort involved in the measurement.

The quantities φ and λ represent the direction vector of the zenith (= true perpendicular direction), which is observed as the point of intersection of the local vertical with the celestial sphere . It results from the precise measurement of star passages or recordings with a zenith camera - initially in the system of star locations , which are then converted into geographical coordinates .

On the other hand, the sizes B and L of the measuring points result from the terrestrial survey network , whose Gauss-Krüger coordinates are converted into latitude and longitude angles on the reference ellipsoid . The deviation from the perpendicular (ξ, η) generally relates to this reference ellipsoid and its fundamental point , which is usually near the center of the country. It is therefore also called the relative deviation from the perpendicular.
If, on the other hand, the national survey is not regional, but relates to the mean earth ellipsoid (e.g. in a GPS survey), then the vertical deviation (ξ, η) is obtained accordingly in a world system , for example the WGS84 . It is then called absolute .

The perpendicular direction is everywhere on the geoid, as this represents the perfect horizontal at sea ​​level . If the earth had no elevations and if it were also completely regular in the interior of the earth , all vertical deviations would be zero and the geoid would be identical to the earth ellipsoid.

Integration of the perpendicular deviation

Every perpendicular direction gives an indication of the small irregularities of the gravitational field and results from the disturbing attraction of the mountains and valleys and their somewhat variable density . The influence of the terrain is up to 50 "in the mountains of Europe, and that of geology around 5–10". This exceeds the modern geodetic measuring accuracy by at least ten times, so that today it has to be taken into account in every precise survey . On the other hand, on plains and in hilly areas, the deviation from the perpendicular varies by only about 5  arc seconds .

To determine the geoid , you not only need the values ​​ξ, η on the measuring points (which are about 10 to 50 km apart, depending on the country), but also their course in between. In the past, they were simply averaged and, at best, the geoid was obtained with a decimeter accuracy. Today, on the other hand, methods of “topographical interpolation ” are used, in which the influence of the mountains is first “deducted” ( topographical reduction ). The smoothed ξ-η values ​​are now interpolated in small steps along the profiles and the terrain is mathematically "added". With this remove-restore process, the approximate vertical deviations are obtained in the entire future geoid network, even where they were not measured at all.

In the last step, these calculated perpendicular deviations are integrated along all profiles, i.e. multiplied by the respective distance. By summation - as with leveling  - one obtains the height differences between the geoid and the earth's ellipsoid in a flat grid , thus completing the geoid determination.

Combined procedures

A combination of different measurement data generally increases accuracy and reliability , but also increases the effort. Combination solutions are particularly effective where methodological weaknesses and strengths are paired.

So can gravimetry or networking with GPS measurements a good altitude deliver the geoid, while the Astrogeoid his inclination best captured. The corresponding combinations are called astro-gravimetric geoid determination or GPS leveling . Some projects in mountainous countries also aim to include measured elevation angles , for which the names L. Hradilek (CZ) and E. Grafarend (D) stand , among others .

Geoid solutions by combining vertical deviation and terrain data were previously called "astro-topographical", but have now become the standard thanks to digital terrain models . An example is the above-mentioned topographical reduction.

Global potential models of the earth can also be combined well with gravimetric or astrogeoids, because they can correct the regional trend of the measurement data when they are reduced. Such harmonic spherical function models have been obtained since the work of RH Rapp and HG Wenzel from the combination of satellite geodesy with terrestrial or flight gravimetry. They are already possible up to degrees and order 720 or 1000, which corresponds to a resolution of 20 to 30 km.

For the oceanic geoid, satellite altimetry with radar is the most important data source because the orbital levels are measured directly above sea level. Because of the ocean currents and tides, which can amount to 1–2 meters in altitude, data models of oceanography are required, which in turn are secured by geodesy at the crossing points of the satellite tracks.

Very promising developments in satellite technology are gradiometry and the SST , with which the special satellites GRACE and GOCE are supposed to record changes in the geoid to a few centimeters. Long-term changes in the earth's gravity field can be recorded in this way, but the spatial resolution is only about 200 kilometers. A combination with more detailed methods is essential here.

Global and regional geoid determination

... ...

History of astro-geodetic geoid determination

See also

literature

Web links