Fly levelling for a canal bank fails despite a clean arithmetic check
A survey party must fix the formation level of a 1.2 km canal bank between two permanent benchmarks: BM-A at chainage 0 with RL 100.000 m, and BM-B at chainage 1200 m with RL 102.400 m. Fly levelling is done in eight instrument setups. In the field book, ΣBS − ΣFS equals the difference between the first and last reduced levels, so the arithmetic check is satisfied. The computed RL of BM-B, however, comes out as 102.462 m. The field notes also show that the total of all backsight distances is 268 m against 772 m of foresight distances, that the last three sights were read near the top of a 4 m staff, and that one turning point was on soft ground where the staff was seen to settle between readings. The specification permits a closing error of ±12√K mm, with K in kilometres. The canal is designed to a bed slope of 1 in 5000, and the tolerance on bed level is ±50 mm.
1. Compute the permissible closing error and rule on whether this survey can be accepted.
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Computation. K = 1.2 km, so √K = 1.0954. Permissible error = ±12 × 1.0954 = ±13.1 mm, say ±13 mm.
Observed misclosure = 102.462 − 102.400 = +0.062 m = 62 mm.
Ratio. 62 / 13.1 ≈ 4.7 times the permitted limit.
Ruling. The survey is rejected. It is not a borderline case that can be argued about: a misclosure five times the specification is a systematic fault, not random reading noise. Random errors in ordinary levelling would be expected to grow with the square root of the distance, which is precisely why the limit is written as 12√K; 62 mm over 1.2 km is far outside that envelope.
What must not happen. Reducing the levels correctly, and even distributing the 62 mm pro rata over the chainage, will produce a neat-looking level book that is still wrong. The book will close perfectly and the bed will still be misplaced. Acceptance depends on the field misclosure, not on the tidiness of the reduction.
2. The arithmetic check passed. Explain why that does not validate the survey, and identify the specific systematic errors visible in the notes.
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Why the check is not a quality check. The identity ΣBS − ΣFS = last RL − first RL is built into the rise-and-fall reduction itself. Any set of readings, however badly taken, will satisfy it if the arithmetic is done consistently. It detects a copying or subtraction slip; it can never detect that the instrument, the staff or the ground was at fault.
Systematic errors visible in the notes:
- Unequal sight lengths - 268 m of backsights against 772 m of foresights. Any residual collimation error (line of sight not parallel to the bubble tube) then accumulates instead of cancelling, and the bias grows with the imbalance.
- Sights near the top of a 4 m staff - maximum reading error and maximum effect of staff tilt; a staff leaning by even 2° over 4 m loses about 2.4 mm on that sight alone, and the error is one-directional if the staffman leans the same way each time.
- Turning point on soft ground - the staff settling between the backsight and foresight at that point directly injects a false rise or fall into the run.
- Long sights - they add refraction and parallax error and make the bubble harder to hold.
The lesson: balance backsight and foresight distances, keep sights short, use a firm turning plate, and check the instrument with a two-peg test before the run.
3. What should the party do next, and under what circumstances would you allow the error to be distributed instead of re-observed? Support the answer with the canal gradient and tolerance.
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Why re-observation is the only honest option here. A 62 mm misclosure spread over 1.2 km is a correction of about 5.2 mm per 100 m, i.e. 0.052 m per km. The canal falls at 1 in 5000, which is 0.20 m per km. The correction would therefore be about 26% of the designed fall - you would be altering the very gradient the survey exists to set, and the bed-level tolerance is only ±50 mm, so a 52 mm/km adjustment would consume the whole tolerance before a single peg is set.
Re-survey plan:
- Service and two-peg test the level; adjust if the collimation error exceeds about 2 mm per 50 m.
- Break the 1.2 km into two loops of about 600 m, each closing on a temporary benchmark, so that any fault is localised.
- Keep backsight and foresight distances within a few metres of each other, sights under about 60 m, staff on a turning plate on firm ground, bubble checked before and after each reading.
- Re-observe until the misclosure is within ±12√K mm (about ±9 mm for 0.6 km).
When distribution is legitimate: only for a small random misclosure within the permissible limit, distributed in proportion to distance (or by the rise-and-fall/height-of-collimation method) purely to make the reduced levels close. A 62 mm systematic error is not that case.