Spirit levelling

Spirit levelling is a technique for deterimining differences in height between points on the Earth's surface. It works by using an instrument, a spirit level, 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 will be marked by markers or benchmarks in the rock or soil. On both of the points, a levelling staff or rod is placed, carrying a numbered graduation 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.

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

For measuring height differences over larger distances, multiple point intervals are chained together, and to eliminate systematics, the reading sequence is "randomized": fb-bf-fb-bf... where fb stands for "forward-backward". In this way, levelling lines[?] are measured which can be interconnected to form a levelling 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, to be used in construction and infrastructural work like civil engineering, one designs and measures a levelling network covering the country to sufficient density. One or more of the nodes of this network should be coastal tide gauge stations. In this way, it becones 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 one such tide gauge or ensemble of tide gauges. Obviously, height datums thus established will differ slightly for different countries as the true sea surface is not a perfect level surface.

If the Earth's gravity field were completely regular and gravity constant, we would always see levelling loops close precisely, That is, we would have

<math> \sum_{i=0}^n \Delta h_i = 0 </math>

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, potential differences do close around loops. We may write

<math> \sum_{i=0}^n \Delta h_i g_i = 0, </math>

where <math>g_i</math> stands for gravity at the levelling interval i. For precise levelling networks on a national scale, always the latter formula should be used, and instead of "height" differences, geopotential differences

<math> \Delta W_i = \Delta h_i g_i </math>

should be used in all computations, producing geopotential values <math>W_i</math> for the benchmarks of the network.

See Physical geodesy for details.