# Leveling

The measurement of height differences between points is called leveling . When geometric leveling the difference in height becomes a horizontally erected leveling read on leveling rods that are placed vertically on the measurement points. With trigonometric leveling , the height differences are calculated from angle and distance measurements. When hydrostatic leveling one is water level gauge used.

Leveling is used in civil engineering, e.g. B. to create flat surfaces, to set up machines horizontally or to determine slopes so that water can flow.

So that a common height reference is given as a starting point for the local leveling over large areas , the state surveying links height points nationwide by leveling to form a height network. This leveling network consists of a coarse-meshed network of very precise fine leveling elements, which are compacted by smaller networks of less precision. This means that even with structures that extend over long distances, such as B. Railway lines, certain heights can be determined.

The leveling methods described here are used to determine height differences between two points and in relation to the gravitational field. Other methods of determining altitude use air pressure ( barometric altitude measurement ), for example , based on stereo image analysis ( photogrammetry ) or satellite geodesy . The heights determined by satellite geodesy are purely geometric and do not relate to the earth's gravity field. Using a laser scanner , three-dimensional coordinates of a surface (e.g. the surface of the earth) and thus also heights can be determined on a massive scale.

## The geometric leveling

The level is set up and leveled at any observation point between the measuring points. In order to eliminate systematic influences such as the residual inclination of the target axis, the curvature of the earth and the refraction , the same target ranges are maintained. A leveling staff is set up vertically at each measuring point . A scale is attached to the leveling staff in such a way that the reading of the staff division in the leveling device gives the vertical distance of the point from the instrument horizon ( device horizon ) of the level. If measurements are taken at another point with the level set up unchanged, the difference between the two readings gives the difference in height between the two points.

The measurement is divided into sections so that a height difference can be measured over a greater distance, over greater height differences or around obstacles. Each section consists of the measurement from the known point to the new point. The leveling device is set up horizontally between the two. This means that the vertical axis is vertical and, if the device is free from defects, the target axis is horizontal.

The reading on the staff at the known point is called backsight , and the reading on the new point is called foresight . The readings are subtracted, backsight minus foresight, to get the elevation gain. A positive difference in height means that the terrain rises in the direction of leveling, a negative one that the terrain falls. When you arrive at the destination, the height differences of all sections are added to obtain the height difference between the starting point and the destination.

With geometric leveling, the metric distance between the point and the device horizon is measured on the measuring stick to determine the height. That is a geometric quantity. So that no water flows between two points, the earth's gravity potential must be the same at both points. The gravity of the earth is a physical quantity. As the length of the leveling train increases, there are therefore deviations between the geometrically determined height and the physical height. Therefore it can happen that water flows between points of the same geometric height.

With the same target ranges, only the curvature of the earth and the direction of gravity are taken into account by leveling the leveling device, but not the magnitude of the acceleration due to gravity . In national surveying, one therefore corrects the heights created by geometric leveling on the basis of gravity measurements .

The standard deviation of a leveling for the German Main Elevation Network 1992 (DHHN92) should not exceed the following values ​​per kilometer of double leveling :

• 1st order leveling net: <1.0 mm
• 2nd order leveling net: <1.5 mm
• 3rd order leveling net: <3.0 mm

## The hydrostatic leveling

The hydrostatic leveling works according to the principle of communicating pipes : If tanks filled with water are connected to each other at the lowest point by pipes, the same water level is established in all tanks. For practical use, a transparent tube is almost completely filled with water without bubbles. If the two ends of the hose are now held approximately at the same height, the water level will be set at both ends of the hose. With hydrostatic leveling with this hose level , there is no line of sight between the measuring points. It is therefore well suited for measurements in buildings. Precision measurement technology offers special attachments (e.g. glass cylinders) with appropriate reading devices or electronic data acquisition for precise reading. This means that hydrostatic leveling is particularly practical for continuous, computer-aided remote monitoring of building movements.

The hydrostatic leveling can bridge distances of up to 20 km between the measuring stations. This allows for leveling trains z. B. wide rivers can be bridged. The reading accuracy is better than 0.02 mm. The overall accuracy is better than 1 mm for distances of a few kilometers. In order to achieve such high accuracy over long distances, however, influences such as B. Temperature differences in the liquid are taken into account during the measurement.

The hydrostatic leveling was already known in ancient times. A trench system was set up around the construction site for the construction of the pyramids . The water level in the trench was the height reference area for the structure.

## Trigonometric level

With trigonometric leveling , the zenith angle z and the inclined distance s ′ to the measuring point are measured with measuring devices ( theodolite , tachymeter, etc.) . The height difference is then calculated as a simple approximation according to the geometric formula or with it . ${\ displaystyle dh = s' \ cdot \ cos \, z}$ ${\ displaystyle dh = s \ cdot \ cot \, z}$ ${\ displaystyle s = s' \ cdot \ sin \, z}$ For target ranges over 200 m, the curvature of the earth and terrestrial refraction must be taken into account.