Reciprocal levelling: why and formula

When and why used: reciprocal levelling is used when the difference of level between two points that are far apart and separated by an obstacle (a wide river, lake, valley) has to be found and the instrument cannot be set midway so that the backsight and foresight are equal. A single setting would include the errors of collimation, curvature and refraction, which are proportional to the sight distance. By taking readings from both banks, these errors cancel.

Derivation: let A and B be the two points, level 1 near A and level 2 near B. Let $e$ be the combined error in the far staff reading (error due to collimation + curvature $-$ refraction), which has the same value and the same sign for both far sights because the distance is the same. The error in the near sights is negligible.

Level near A: staff readings $a_1$ (near, on A) and $b_1$ (far, on B). Level near B: $a_2$ (far, on A) and $b_2$ (near, on B).

First set:

$$H_1 = a_1 - (b_1 - e) = (a_1 - b_1) + e$$

Second set:

$$H_2 = (a_2 - e) - b_2 = (a_2 - b_2) - e$$

The true difference H is the same in both, so $H_1 = H_2$:

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

Adding $H_1$ and $H_2$ gives:

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

So the true difference of level is the mean of the two apparent differences. $H$ is the rise of B above A (BS $-$ FS is a rise), so RL of B $=$ RL of A $+ H$; a negative $H$ means B is lower than A.

Best method: the instrument is set near each bank in turn and readings are taken on the two staffs with the same instrument, the same staff, and the same observer, in as short a time as possible, and the mean taken of several sets. The errors removed are:

1. Error due to collimation (line of sight not horizontal),
2. Error due to curvature of the earth, and
3. Error due to refraction of the atmosphere, assuming that it is the same for the two sets.