Spirit leveling

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Spirit leveling is a technique for determining differences in height between points on the Earth's surface. It works by using a spirit level, an instrument consisting of a telescope and a tube level like that used by carpenters, rigidly connected. When the bubble in the tube level is in the middle, the telescope's optical axis (collimation axis) will point exactly in the direction of the local horizontal.

The spirit level is placed on a tripod in the middle between the two points whose height difference is to be determined; the points are marked by markers or benchmarks in the rock or soil. A leveling staff or rod is placed on each point, with measured graduations, usually in centimetres and fractions thereof. The observer focuses in turn on each rod and reads the value from it. Subtracting the "back" and "forward" value provides the height difference.

For the greatest precision the distances to the rods should not be too large, typically 30-60 m, and should be approximately equal in order to eliminate systematic errors such as the residual misalignment between telescope axis and tube level axis.

For measuring height differences over larger distances, multiple point intervals are chained together, and to eliminate systematic errors the reading sequence is "randomized": fb-bf-fb-bf... where fb stands for "forward-backward". In this way, leveling lines are measured that can be interconnected to form a leveling network. The closure of the loops of such a network provide a way to check the correctness of the measurements.

To establish a height system for a country or area for use in construction and infrastructural work like civil engineering, a leveling network covering the country to sufficient density is designed and measured. One or more of the nodes of this network should be coastal tide gauge stations. In this way, it becomes possible to obtain the heights of all the points in the network in a system, or height datum, the zero point of which is given by mean sea level at a tide gauge or ensemble of tide gauges. Height datums thus established will differ slightly between different countries, as the sea surface is not a perfect level surface.

If the Earth's gravity field were completely regular and gravity constant, leveling loops would always close precisely:

$\sum_{i=0}^n \Delta h_i = 0$

around a loop. In the real gravity field of the Earth, this happens only approximately; on small loops, the loop closure is negligible, but on larger loops it is not.

Instead of height differences, geopotential differences do close around loops:

$\sum_{i=0}^n \Delta h_i g_i = 0,$

where $g i$ stands for gravity at the leveling interval i. For precise leveling networks on a national scale, the latter formula should always be used.

$\Delta W_i = \Delta h_i g_i\,$