Height measurement
The height measurement is a branch of Geodesy , partly also the geography (topography) and the machine or construction technology. Heights are required and determined in a variety of areas, e.g. B. in engineering and land surveying , in geology and spatial planning, in mechanical engineering and construction , in mountaineering or in navigation .
Occasionally one speaks of altimetry instead of height measurement  especially when it comes to translations from English, altimeter instruments and satellite altimetry .
Delimitations
Height measurements can have very different reference surfaces, which must be taken into account when comparing data  see height (geodesy) :
 "Relative heights" are a measure of how much an object towers over the surroundings (important for example in construction, in aviation, in mountain sports and for calculating energy consumption).
 “Absolute heights” are physical and relate to the earth's gravity field and therefore, in practice, to the mean sea level ( geoid ) as a horizon reference . The state height networks are, however, defined by different levels and not processed uniformly, which can lead to differences in the cm to dm range.
 "Heights from satellite measurements" are purely geometric and refer to a mathematical earth ellipsoid (today mostly WGS84 system). The difference to sea levels can reach 100 meters (in Central Europe approx. 50 m, see geoid ). Because of the geometrical nature of these measurements, “water can flow between points of the same GPS height” (W. Torge 1999), so that they need a correction for surveying purposes.
Methods of height measurement
The height measurement itself can be done in very different ways:

leveling (with respect to a level surface of the earth):
 by measuring geometric height differences with the leveling device
 by measuring physical height differences with the hose level
 trigonometric  by measuring elevation angles and inclined distances (see trigonometric leveling , tachymetry and instrument height )
 using GPS or GNSS satellites (see ellipsoidal altitude and GPS leveling )
 by measuring the time of flight of sound waves in water, for example with an echo sounder
 by measuring the time of flight of microwaves with continuous wave radar

barometric  by measuring air pressure differences with an aneroid barometer ( hypsometer )
 barometric altitude measurement in geodesy
 Altimeter for cartography and hiking
 barometric altitude in aviation , recording the heights with height Schreiber
 gravimetric  with a gravimeter (→ dynamic height )
 photogrammetrically  by evaluating stereo image pairs from the air
 by means of remote sensing , e.g. B. by measuring the transit time of laser or radar waves (→ radar interferometry )
 from the air (→ laser scanner )
 from space (→ satellite altimetry , laser methods of satellite geodesy , SRTM radar).
Special measuring methods and height marks
In addition, special measuring methods in mechanical engineering and laboratory technology should be mentioned, for example when adjusting machine axes or measuring the level of liquids. Geodesy also knows the astronomical level for the precise analysis of the earth's gravity field (see geoid determination ) and the measurement of space polygons , in which position and height measurements are combined.
The accuracy ranges from hundredths of a millimeter for precision leveling to millimeters for trigonometric height measurement and centimeters for GPS to a decimeter or a few meters for satellite altimetry and barometric height measurement. The fixed points of the precision leveling system must be marketed in a particularly stable manner because of their high accuracy, as they are not only the basis of all technical leveling systems, but also for the vertical movements of the earth's crust , which, depending on the geology of the subsurface, amount to 0.01 mm to a few millimeters per year. The height markers are therefore carefully walled into old buildings that are no longer subject to settlement , such as churches (see tower bolts ) and official buildings (see picture), or attached in natural rock or on deep foundations.
The strict definition of “height” is a multifaceted problem because it can be done in a number of ways, such as: B. purely geometrically or gravimetrically physically . Subjects involved are higher geodesy , geometry and potential theory .
Relative height measurements with little effort
 It is not always possible to use more effort (equipment and / or arithmetic operations) to determine the height of an object.
 The achievable accuracies are limited as a result, but are sufficient for many applications (e.g. tree height measurements).
 The most common methods are based on angle measurements with a clinometer and an isosceles right triangle.
 The mirror hypsometer is also based on angle measurements .
 A method that is not based on direct angle measurements is called height measurement with the forester's stick, which always requires an assistant for measurement and is usually limited to tree height measurements.
According to the mathematical principle that to calculate an unknown quantity in a triangle, two additional quantities are always required, the quantities distance to the measurement object ("base line") and the "elevation angle" (angle at which the highest point to be measured are used) are used for height measurement can be seen at the end of the baseline) for the calculation.
If a clinometer with graduation is used, the height of the object in m results from the tangent relationship ( angle function ):
 Baseline (m) × tan elevation angle = object height
A table or suitable calculator / slide rule is required.
If a clinometer with a% scale is used, the height of the object in m results from the formula:
 % Value / 100 × baseline (m)
The measurement with the triangle is based on the fact that the tangent of an angle of 45 ° = 1. It follows from this that the height of the object is equal to the baseline if the highest point of the object is seen at an angle of 45 °. This is the case when the object is aligned using the hypotenuse of the triangle (which can be folded out of paper) (the cathetus of the triangle pointing to the ground must be kept horizontal ).
Since the measurements described above are determined from "eye level", this height must be added to the result (for example 1.70 m).
literature
 Wolfgang Torge : Geodesy. 3rd, completely revised and extended edition. de Gruyter, Berlin et al. 2001, ISBN 3110170728 .
 Heribert Kahmen : Surveyors. 18th, completely revised and expanded edition. de Gruyter, Berlin et al. 1993, ISBN 311013733X .