Reciprocal Levelling

Let the level difference among stations A and B is required to be measured precisely, and it is not probable to set up instrument midway among A and B (Figure 8). At first, instrument is set up and levelled extremely near to A and staff is held at A and B and readings recorded correspondingly. The instrument is then shifted to a position O₂ very near to station B, and staff readings once again taken for station A and B correspondingly.

Let a₁ and b₁ be the readings from position O₁, while a₂ and b₂ are readings along with instrument at O₂. If there is no collimation error, the line of collimation should coincide along with horizontal line. If there is no error due to curvature and refraction etc., the level line should merge within horizontal line. Therefore, in an error free environment there shall be a single line in place of three, both in instrument position at O₁ (Figure 8(a)) and at O₂ (Figure (b)).

                                              Figure : Reciprocal Levelling

Since instrument position O₁ is extremely near to or exactly at A, there will be practically no deviation in staff reading at A (i.e. reading a1 is correct) although the true reading (level difference among A and B) would be

d = (b₁ - e) - a₁                                                                                                          . . . (4.2)

Similarly for instrument position extremely near to or exactly at B, the true level difference would be

d = b₂ - a₂ + e                                                                                                            . . . (4.3)

Adding Eqs. (4.2) and (4.3), we get

2d = (b₁ - a₁) + (b₂ - a₂) or d = 1/2 {(b₁ - a₁) + (b₂ - a₂)}                                       . . . (4.4)

Eq. (4.4) does not have a "e" term hence is error free. The magnitude of error e is acquired through subtracting Eq. (4.3) from Eq. (4.2).

i.e.       e = 1/2 {(b₁ - a₁) - ( b₂ - a₂)}

This procedure of reciprocal levelling eliminates errors because of collimation and curvature. The refraction error might not be fully eliminated as there is a possibility that refraction of the air might change during shifting of instrument from O₁ to O₂. Therefore for more accurate results and to eliminate any probable refraction error, two independent instruments are set at stations O₁ and O₂ concurrently and readings a₁, a₂, b₁ and b₂ recorded at the similar instant.