Perpendicular deviation

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Earth's gravitational field : vertical direction, geoid and equipotential
Vertical deviation: difference between the true vertical direction and a theoretical earth ellipsoid . It causes an apparent displacement and influences all terrestrial direction measurements

The perpendicular deviation is the angle between the perpendicular direction and the ellipsoid normal at a certain point on the earth. It can reach 30 to 50 arc seconds (approx. 0.01 °) in high European mountains  , less in the lowlands, and corresponds to the local inclination of the geoid to the ellipsoid of revolution of the national survey .

Even if the vertical deviation is zero, the vertical direction (implemented e.g. by a freely hanging plumb line) does not point to the center of the earth, but rather up to about 700 arc seconds (0.2 °) because of the flattening of the earth (centrifugal force of the earth's rotation )  .

Sometimes in the construction industry one speaks of deviation from the perpendicular when a building or facade is out of perpendicular .


According to the dictionary of the Fédération Internationale des Géomètres , the perpendicular deviation is the angle between the perpendicular direction at a point and the normal assigned to this point by a projection on an ellipsoid of revolution.

One speaks of an astro- geodetic deviation from the perpendicular if the direction of the perpendicular was determined using the methods of geodetic astronomy. It occurs when transforming between local coordinate systems . In contrast, the gravimetric deviation from the perpendicular is based on the determination of the perpendicular direction through gravity measurements and is obtained from the solution of the geodetic boundary value problem .

Deviations from the perpendicular depend on the ellipsoidal coordinates and thus on the parameters of the reference ellipsoid and its position in relation to the earth. If the reference ellipsoid is a geocentric (in the earth's center of gravity ) and at the same time a mean earth ellipsoid , one speaks of absolute vertical deviations, otherwise relative vertical deviations.


The amount that the deviation from the perpendicular can reach depends on several factors:

  1. on the topography - the height and roughness of the terrain. In the Alps , individual mountain ranges can cause local deflections in the vertical direction of up to almost 60 arc seconds (approx. 0.015 °).
  2. on geology - the subterranean course of the rock layers . Where the horizontal storage is severely disturbed - such as B. in Molasse basin or in the Swiss Ivreazone - even regional distractions over 60 arc seconds are possible.
  3. of the storage of the reference ellipsoid of the national survey - see also geodetic datum .

While in the high mountains of Europe the mean vertical deviations remain below 30  arc seconds (the maximum values ​​can reach around 60 arc seconds), in the Andes and Himalayas almost double the amounts are possible.

Given the accuracy of modern measurements , the deflection of the vertical in almost every project or survey network affects as soon as the sightings (line of sight) differ by more than a few degrees from the horizontal. The effects must therefore usually be reduced mathematically , which is the subject of astrogeodesy and higher geodesy ( earth measurement ).

In the hill country these effects reach a few arc seconds or a few centimeters per kilometer, in the mountains up to ten times that. That z. B. the earlier tunnels still fit together relatively precisely, is due to the approximate symmetry of most mountain ranges.


Extensive measurements of the vertical deviation were first carried out around 1800 after theoretical investigations by Carl Friedrich Gauß in the course of the Hanoverian land surveying, in the area of ​​the Harz , where Gauß and his assistants expected the greatest effects. In the 1970s, the TU Hannover under Wolfgang Torge established a modern astro-geodetic test network West Harz .

But such considerations and suitable astrogeodetic measurements by the Scottish researchers James Hutton and Nevil Maskelyne had already been made 20 years earlier . In order to determine the rock density of the Shehellien Mountains, they selected measuring points on both sides and compared their (then still laborious) measured distance with the difference between their astronomically measured latitudes . The difference in angle was 11.6  arc seconds and the rock density was 2.6 to 2.8 g / cm³.

In the early 19th century, the Indian land survey under George Everest showed that the vertical deviations are particularly large in the Himalayas . Nevertheless, the observations resulted in significantly smaller values ​​than calculated from the mountain masses. Scientists Airy and Pratt explained this around 1855 by a mass compensation in the lower crust of the earth, which led to two theories of isostasis .

While the north-south component of the vertical deviation (or the astronomical latitude ) could already be measured over 200 years ago, the east-west component requires an exact astronomical longitude determination and therefore a precise time system. Such has only been available since the invention of radio technology and the subsequent establishment of world time , which is now distributed by time signal transmitters . It was therefore only in the 20th century that it became possible to determine the deviation from the perpendicular on a larger scale.

Around 1930, astro-geodetic measurements became a standard method of geodesy and the most important basis for the required astro-geodetic network adjustment , as the accuracy of the surveying networks no longer met the growing needs. From the 1950s onwards, it was possible to accelerate the previously complex plumb line measurements by means of semi-automatic angle and time registration. Between 1970 and 2000, research on the subjects of plumb bias, geoid and geodetic gravimetry reached a climax due to four current requirements at the same time :

  1. the absolute need for surveying networks with accuracies better than 1: 1,000,000 (mm per km),
  2. the increasing construction of road tunnels through the Alps and other mountains, where the vertical deviation sometimes amounts to a few decimeters per kilometer,
  3. the demand for the so-called centimeter geoid (term first coined by Torge , see below), because as early as 1980 the appearance of satellite positioning with centimeter accuracy ( GPS , GLONASS , SLR ) and cosmic interferometry ( VLBI ) could be foreseen,
  4. the need for potential theoretical investigations of the earth's crust , for which the vertical deviation can provide better geological layer inclinations than conventional gravimetry - see z. For example, today's major German project " Sedimentary Basin ", the investigations by Gerstbach (TU Wien) and Papp in the Vienna Basin, by Gurtner ( Ivrea body in southern Switzerland ) and the extensive TESLA project planning for the 30 km linear accelerator near Hamburg.

Through various large-scale projects in Central Europe (especially Germany, Austria, Switzerland, as well as Slovenia and Slovakia), in Southern Europe (Croatia, Greece, Turkey) and in South America (especially Argentina), the geoid in the German-speaking area has been reduced to 2 up to 5 cm, elsewhere improved to 5 to 10 cm. In Germany, the " astrogeoid " determined from vertical deviations competes with the "gravimetric geoid", while mountainous countries such as Austria, Switzerland, Slovakia and Greece prefer the astrogeoid. A dense network of thousands of vertical deviation points and hundreds of Laplace points has been available in these countries since around 1990 (point spacings between 7 and 10 km on the one hand and 50 km on the other hand), around the world there are tens of thousands of measurement points where the exact vertical direction is on the surface of the earth was measured.