Reciprocal Levelling Across the Budhi Gandak
A level and staff are used to transfer a bench-mark level across the Budhi Gandak near a bridge site in Begusarai district. The river is about 180 m wide, so the instrument cannot be set midway. Peg A is on the west bank and peg B on the east bank. With the level set close to A, the backsight on A is 1.462 m and the foresight on B is 2.318 m. With the level set close to B, the backsight on B is 1.225 m and the foresight on A is 2.095 m. A direct differential levelling run along a detour gives B lower than A by 0.842 m. The staff is a 5 m telescopic staff, and the bubble is in adjustment. The assistant proposes to accept the detour value because it is a straight calculation; the site engineer wants the reciprocal result. The reduced level of A is 48.620 m.
1. Compute the true difference in level between A and B from the reciprocal readings and the reduced level of B. Compare with the detour value.
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Reciprocal computation
- Level near A: $\Delta h_1=BS_A-FS_B=1.462-2.318=-0.856$ m.
- Level near B: $\Delta h_2=BS_B-FS_A=1.225-2.095=-0.870$ m.
- Mean: $(-0.856-0.870)/2=-0.863$ m. B is 0.863 m lower than A.
- $RL_B=48.620-0.863=47.757$ m.
The detour gives $48.620-0.842=47.778$ m. The two values differ by 0.021 m (21 mm). Adopt the reciprocal value for the bench mark because the method is designed to cancel the long-sight errors; use the detour as an independent check, not as the primary result. A 21 mm disagreement is a warning to check the detour closure and staff zero, not a reason to average blindly.
2. Why do the two reciprocal values differ from the detour? Which instrumental and atmospheric errors cancel, and which do not? Is 21 mm significant for a 1:500 site plan with a 0.25 m contour interval?
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Why they differ
The two reciprocal setups reverse which sight is long. A collimation error (line of sight not parallel to the bubble tube) adds the same amount to a long sight in one setup and subtracts it in the other, so it largely cancels in the mean. Curvature and refraction also act on the long sight and largely cancel for the same reason. What does not cancel: staff zero error, bubble/level error, non-vertical staff, reading and plumbing errors, and any change in the instrument between setups.
Significance
At 1:500, 21 mm is 0.042 mm on paper, far below plotting resolution. With a 0.25 m contour interval, 21 mm is 8.4% of the interval, so it will not change the shape of contours drawn from spot levels. It is still material for a bench mark, a drainage invert, or a floor level. Use the reciprocal value and close the run on a second bench mark; do not use will-not-show-on-the-map as the only justification.
3. List the field checks and record entries you would require before signing the level book.
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Field checks
- Confirm the bubble is centred and the instrument is on stable ground; keep the staff vertical, preferably with a circular bubble.
- Keep ordinary sight lengths balanced where possible; for the reciprocal sets, record the distance to the near and far staff.
- Close the level run on a second bench mark; compare the observed and known reduced level.
- Check the staff zero and joint; check that the 5 m staff is not read above its graduations.
Record entries
Instrument and staff numbers, date and time, observer, station descriptions, near/far distances, all four readings, computed $\Delta h$ for each setup, mean, adopted reduced level, detour value and closure. Note that the reciprocal value governs and why.