Principles of levelling; reciprocal levelling

Basic principles of levelling

1. A horizontal line of sight is established with a level (the line of collimation is made horizontal by centring the bubble).
2. The vertical distance from this line of sight to the points whose elevations are required is measured with a graduated levelling staff held vertically on them.
3. The difference of staff readings gives the difference of level between two points: the point with the smaller reading is higher.
4. If the RL of one point is known, the RL of the other is: RL = known RL $\pm$ difference of level, usually found through the height of instrument or the rise and fall method.
5. When the points are far apart or at different heights, the work is carried out in steps using change points; sights are balanced to remove the errors of collimation, curvature and refraction.

Reciprocal levelling

Used when two points A and B are on opposite sides of a river or valley and the level cannot be set at the middle.

   Set 1: level near A            Set 2: level near B
   A [L] ------------> B          A <------------ [L] B
   a1(near)      b1(far)          a2(far)      b2(near)

1. Set the level near A; read the staff at A ($a_1$) and at B ($b_1$).
2. Move the level near B; read the staff at A ($a_2$) and B ($b_2$).
3. The true difference of level is the mean of the two apparent differences:

$$H = \frac{(a_1 - b_1) + (a_2 - b_2)}{2}$$

The error in the far reading (collimation, curvature, refraction) is

$$e = \frac{(a_2 - b_2) - (a_1 - b_1)}{2}$$

The mean eliminates these errors because the far sight is of the same length in each set. The two sets should be taken at about the same time under the same atmospheric conditions, and the mean of several sets is used for precise work.