Leveling Link for Settlement Monitoring: Closing the Loop on a Systematic Misclosure
A 12 km leveling link between two national benchmarks is run with a digital level and invar staffs to provide vertical control for embankment settlement monitoring on a new rail line. BM-A has an adopted height of 24.316 m and BM-B of 31.902 m; the measured height difference is +7.604 m. The link comprises 96 setups, and the summed backsight-minus-foresight sight-length imbalance across the line is +1,200 m. A two-peg test on the same instrument indicates a collimation error of +0.015 mm per metre of sight-length imbalance. The project specification for the link is a misclosure not exceeding 8 mm√K, where K is the length in kilometres. The settlement monitoring itself requires 2 mm accuracy at 95% at individual points. Field notes record one turning point on soft ground, and many readings taken at midday across hot ballast.
1. Compute the misclosure and the allowable value for this link, and state whether it passes.
Read the worked answer
Measured versus known height difference:
- Known: 31.902 − 24.316 = +7.586 m
- Measured: +7.604 m
- Misclosure = 7.604 − 7.586 = +0.018 m (18 mm)
Allowable: 8 mm√K with K = 12 km:
- √12 = 3.464
- Allowable = 8 × 3.464 = 27.7 mm
Verdict: 18 mm < 27.7 mm, so the link passes the link specification by that criterion alone.
But passing is not the same as being usable. The 2 mm requirement for individual settlement points is an order of magnitude tighter than the link result. A 18 mm misclosure distributed over 96 setups places roughly 18 mm of uncertainty at the far end of the link, before any monitoring error is added. If the monitoring network is referenced to a point at the BM-B end, every settlement reading inherits a systematic offset an order of magnitude larger than the movement being measured - which would make the monitoring programme worthless regardless of how precise each individual reading is. The link spec is the wrong test; the monitoring spec governs.
2. Test the hypothesis that the misclosure is caused by the collimation error rather than by random error.
Read the worked answer
The arithmetic matches almost exactly.
Collimation error contribution = error per unit imbalance × total imbalance
- = 0.015 mm/m × 1,200 m
- = 18.0 mm
Observed misclosure = 18 mm. The agreement is essentially exact.
What this means:
- The misclosure is systematic, not random. Random error over 96 setups with sub-millimetre readings would scatter by a few millimetres; 18 mm is far outside that.
- Therefore it must not be distributed by an equal or Bowditch-style adjustment across the setups and then forgotten. Distributing a known systematic error makes every adjusted height individually plausible, hides the cause, and leaves the instrument still out of adjustment for the next campaign.
- The mean imbalance is 1,200/96 = 12.5 m per setup - sight lengths were systematically unequal, which is exactly the condition that lets collimation error into a link line. This is a field-method failure, not an instrument-only failure.
Counter-check: if the collimation test result were wrong, the discrepancy should not be reproducible. Re-run the two-peg test and repeat two or three setups with deliberately unequal sights (say 5 m vs 45 m) and confirm the recoverable offset; that distinguishes a genuine collimation error from a staff zero error, which would produce a constant offset proportional to the number of staff placements rather than to imbalance.
3. What must be done before the leveling is accepted for settlement monitoring, and what field practice must change?
Read the worked answer
Do not accept these heights. Re-observe the link after the following.
Instrument and calibration:
- Re-adjust the collimation (the cross-hair or the compensator) until the two-peg test gives a residual below about 0.005 mm/m, and record the test sheet with every campaign.
- Verify the invar staffs against a certified baseline; check the staff bubble and the zero mark.
Field method:
- Balance sight lengths to within ±2–5 m per setup. With 96 setups, a 5 m mean imbalance and 0.005 mm/m residual error gives about 2.4 mm total - acceptable. The current 12.5 m mean is not.
- Replace the soft-ground turning point with a driven steel peg or a stable plate; a settling tripod foot produces a one-off blunder that no adjustment can detect.
- Avoid midday readings over hot ballast. Refraction over a heated surface bends the line of sight upward and varies unpredictably; read in the early morning or under cloud, and keep the line of sight at least 0.5 m above the ground.
- Use the same instrument, staffs and observers across all monitoring epochs - change them and the systematic errors change with them, corrupting the comparison.
- Reference the monitoring network to a deep benchmark outside the settlement influence zone, and run closed loops so that closures are self-checking.
Acceptance: re-run the link, report misclosure against 8 mm√K, and separately demonstrate 2 mm at 95% by repeat observation of a stable test point.