### Spirit leveling

For other uses, see Levelling (disambiguation).
"Single-levelling" redirects here. For other uses, see Single-level (disambiguation).

Levelling (or leveling in American spelling) is a branch of surveying, the object of which is to

1. Find the elevation of a given point with respect to the given or assumed Datum.
2. Establish a point at a given elevation with respect to the given or assumed Datum.

Levelling is the measurement of geodetic height using an optical levelling instrument and a level staff or rod having a numbered scale. Common levelling instruments include the spirit level, the dumpy level, the digital level, and the laser level.

## Spirit (Optical) levelling

Spirit levelling employs a spirit level, an instrument consisting of a telescope with a crosshair and a tube level like that used by carpenters, rigidly connected. When the bubble in the tube level is centered the telescope's line of sight is supposed to be horizontal (i.e. perpendicular to the local vertical).

The spirit level is on a tripod midway between the two points whose height difference is to be determined. A leveling staff or rod is held vertical on each point; the rod is graduated in centimetres and fractions or tenths and hundredths of a foot. The observer focuses in turn on each rod and reads the value. Subtracting the "back" and "forward" value provides the height difference.

We can't expect the instrument to be in perfect adjustment, but we can hope that when the bubble is centered the telescope's line of sight is always the same small angle off of horizontal. If it is, we can still level accurately by setting the instrument equidistant from the points to be measured, so the errors cancel.

## Leveling Procedure

A typical procedure is to set up the instrument within 100 metres (110 yards) of a point of known or assumed elevation. A rod or staff is held vertical on that point and the instrument is used manually or automatically to read the rod scale. This gives the height of the instrument above the starting (backsight) point and allows the height of the instrument (H.I.) above the datum to be computed.

The rod is then held on an unknown point and a reading is taken in the same manner, allowing the elevation of the new (foresight) point to be computed. The procedure is repeated until the destination point is reached. It is usual practice to perform either a complete loop back to the starting point or else close the traverse on a second point whose elevation is already known. The closure check guards against blunders in the operation, and allows residual error to be distributed in the most likely manner among the stations.

Some instruments provide three crosshairs which allow stadia measurement of the foresight and backsight distances. These also allow use of the average of the three readings (3-wire leveling) as a check against blunders and for averaging out the error of interpolation between marks on the rod scale.

The two main types of levelling are single-levelling as already described, and double-levelling (Double-rodding). In double-levelling, a surveyor takes two foresights and two backsights and makes sure the difference between the foresights and the difference between the backsights are equal, thereby reducing the amount of error. Double-levelling costs twice as much as single-levelling.

## Refraction and Curvature

The curvature of the earth means that a line of sight that is horizontal at the instrument will be higher and higher above a spheroid at greater distances. The effect may be significant for some work at distances under 100 meters.

The line of sight is horizontal at the instrument, but is not a straight line because of refraction in the air. The change of air pressure with elevation causes the line of sight to bend toward the earth. The amount of refraction depends slightly on air temperature and pressure.

The combined correction is approximately:

$\Delta h_\left\{meters\right\} = 0.067 D_\left\{km\right\} ^2$ or $\Delta h_\left\{feet\right\} = 0.021 \left\left(\frac \left\{D_\left\{ft\right\}\right\}\left\{1000\right\} \right\right)^2$

For precise work these effects need to be calculated and corrections applied. For most work it is sufficient to keep the foresight and backsight distances approximately equal so that the refraction and curvature effects cancel out.

## Leveling loops and Gravity Variations

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

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

around a loop. In the real gravity field of the Earth, this happens only approximately; on small loops typical of engineering projects, the loop closure is negligible, but on larger loops covering regions or continents it is not.

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

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

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\$

should be used in all computations, producing geopotential values $W_i$ for the benchmarks of the network.

## Levelling Instruments

### Older Instruments

The wye level is the oldest and bulkiest of the older style optical instruments. A low-powered telescope is placed in a pair of clamp mounts, and the instrument then leveled using a spirit level, which is mounted parallel to the main telescope.

The dumpy level was developed by English civil engineer William Gravatt, while surveying the route of a proposed railway line form London to Dover. More compact and hence both more robust and easier to transport, it is commonly believed that dumpy levelling is less accurate than other types of levelling, but such is not the case. Dumpy levelling requires shorter and therefore more numerous sights, but this fault is compensated by the practice of making foresights and backsights equal.

Precise Level designs were often used for large leveling projects where utmost accuracy was required. They differ from other levels in having a very precise spirit level tube and a micrometer adjustment to raise or lower the line of sight so that the crosshair can be made to coincide with a line on the rod scale and no interpolation is required.

### Automatic level

Automatic levels make use of a compensator that ensures that the line of sight remains horizontal once the operator has roughly leveled the instrument (to within maybe 0.05 degree). The surveyor sets the instrument up quickly and doesn't have to relevel it carefully each time he sights on a rod on another point. It also reduces the effect of minor settling of the tripod to the actual amount of motion instead of leveraging the tilt over the sight distance. Such levels became standard in the later part of the twentieth century. Three level screws are used to level the instrument.

### Digital Level

Digital levels electronically read a bar-coded scale on the staff. These instruments usually include data recording capability. The automation removes the requirement for the operator to read a scale and write down the value, and so reduces blunders. It may also compute and apply refraction and curvature corrections.

### Laser level

Main article: Laser level

Laser levels  project a beam which is visible and/or detectable by a sensor on the leveling rod. This style is widely used in construction work but not for more precise control work. An advantage is that one person can perform the levelling independently, whereas other types require one person at the instrument and one holding the rod.

The sensor can be mounted on earth-moving machinery to allow automated grading.