Reciprocal levelling and its significance

Reciprocal levelling is the method of finding the difference of level between two points, such as the banks of a wide river or a valley, where it is not possible to set the instrument at the mid-point. Two sets of readings are taken, one with the instrument close to each point.

Procedure

1. Set up the level near A (on one bank), and take staff readings on A (near, $a_1$) and B (far, $b_1$).
2. Shift the level close to B (other bank) and take readings on B (near, $b_2$) and A (far, $a_2$).
3. Use the same instrument and staff, and keep the time between the two sets short, so that the atmospheric conditions are the same.

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

Significance

• A single setting with unequal sight lengths contains the errors due to collimation (line of collimation not parallel to the axis of the bubble tube), curvature of the earth and atmospheric refraction. The collimation error grows in proportion to the sight distance, and the curvature and refraction errors in proportion to its square, so they are large for the long sight across a river.
• In reciprocal levelling the far sight in each set is the same length, so the errors have the same magnitude and opposite effects on the two differences; they cancel in the mean:

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

• It gives the correct difference of level without balancing sights and without measuring the width of the river.

Check on precision

The two apparent differences should agree. The difference between them, $(a_1 - b_1) - (a_2 - b_2)$, equals twice the combined error in the far reading. If it is within the permissible error of the work (a few millimetres for a short crossing; the allowable closing error of $\pm 24\sqrt{K}$ mm, with K in km, for ordinary levelling is a guide), the work is acceptable. If it is large, the observations are repeated, preferably at a time of steady atmosphere (early morning or late afternoon).