Levelling

Stadia marks on a crosshair while viewing a metric levelling rod. The top mark is at 1500 mm and the lower is at 1345 mm. The distance between the marks is 155 mm, yielding a distance to the rod of 15.5 m.

Optical levelling employs an optical level, which consists of a precision telescope with crosshairs and stadia marks. The cross hairs are used to establish the level point on the target, and the stadia allow range-finding; stadia are usually at ratios of 100:1, in which case one metre between the stadia marks on the levelling staff represents 100 metres from the target. The complete unit is normally mounted on a tripod, and the telescope can freely rotate 360° in a horizontal plane. The surveyor adjusts the instrument's level by coarse adjustment of the tripod legs and fine adjustment using three precision levelling screws on the instrument to make the rotational plane horizontal. The surveyor does this with the use of a bull's eye level built into the instrument mount. The surveyor looks through the eyepiece of telescope while an assistant holds a vertical level staff which is a graduated in inches or centimeters. The level staff is placed vertically using a level, with its foot on the point for which the level measurement is required. The telescope is rotated and focused until the level staff is plainly visible in the crosshairs. In the case of a high accuracy manual level, the fine level adjustment is made by an altitude screw, using a high accuracy bubble level fixed to the telescope. This can be viewed by a mirror whilst adjusting or the ends of the bubble can be displayed within the telescope, which also allows assurance of the accurate level of the telescope whilst the sight is being taken. However, in the case of an automatic level, altitude adjustment is done automatically by a suspended prism due to gravity, as long as the coarse levelling is accurate within certain limits. When level, the staff graduation reading at the crosshairs is recorded, and an identifying mark or marker placed where the level staff rested on the object or position being surveyed.

Diagram showing relationship between two level staff, or rods, shown as 1 and 3. The level line of sight is 2.

A typical procedure for a linear track of levels from a known datum is as follows. 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 difference between these two readings equals the change in elevation, which is why this method is also called differential levelling. 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.[1] Double-levelling costs twice as much as single-levelling.[2]

The other standard method of levelling in construction and surveying is called trigonometric levelling, which is preferred when levelling "out" to a number of points from one stationary point. This is done by using a total station, or any other instrument to read the vertical, or zenith angle to the rod, and the change in elevation is calculated using trigonometric functions (see example below). At greater distances (typically 1,000 feet and greater), the curvature of the Earth, and the refraction of the instrument wave through the air must be taken into account in the measurements as well (see section below).

Ex: an instrument at Point A reading to a rod at Point B a zenith angle of < 88°15'22" (degrees, minutes, seconds of arc) and a slope distance of 305.50 feet not factoring rod or instrument height would be calculated thus:

cos(88°15'22")(305.5)= 9.30 ft.,

meaning an elevation change of approximately 9.30 feet in elevation between Points A and B. So if Point A is at 1,000 feet of elevation, then Point B would be at approximately 1,009.30 feet of elevation, as the reference line for zenith angles is straight up going clockwise one complete revolution, and so an angle reading of less than 90 degrees (horizontal or flat) would be looking uphill and not down (and opposite for angles greater than 90 degrees), and so would gain elevation.

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 insignificant 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 atmospheric refraction. The change of air density with elevation causes the line of sight to bend toward the earth.

The combined correction for refraction and curvature is approximately:[3]

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

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. Refraction is generally the greatest source of error in leveling. For short level lines the effects of temperature and pressure are generally insignificant, but the effect of the temperature gradient dT / dh can lead to errors.[4]

Assuming error-free measurements, if the Earth's gravity field were completely regular and gravity constant, leveling loops would always close precisely:

${\displaystyle \sum _{i=0}^{n}\Delta h_{i}=0}$

around a loop. In the real gravity field of the Earth, this happens only approximately; on small loops 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:

${\displaystyle \sum _{i=0}^{n}\Delta h_{i}g_{i},}$

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

${\displaystyle \Delta W_{i}=\Delta h_{i}g_{i}\ }$

should be used in all computations, producing geopotential values ${\displaystyle W_{i}}$ for the benchmarks of the network.

• Glossary of levelling terms

• Survey camp
• Land levelling
• Orthometric height
• Dynamic height
• Physical geodesy