A Level Loop That Passes the Test but Fails the Reasonableness Check
You are running a 5.2-mile closed differential leveling loop for a drainage design: 44 setups, a 32× automatic level, and a 25-ft fiberglass rod with a circular bubble. The loop miscloses +0.086 ft. The project specification requires third-order work, taken as 0.05 ft times the square root of M, with M in miles. A two-peg test on the level shows the line of sight rises 0.011 ft per 100 ft of sight distance. Reviewing the field book, backsight and foresight distances were paced but not recorded; the rodperson recalls that setups were usually on the uphill side with the backsight consistently longer than the foresight by roughly 10 ft. The rod bubble was cracked and taped in place for the last 12 setups. The loop is due for adjustment and submittal on Friday.
1. Does the loop meet the specification? What else must be checked before the misclosure is accepted?
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The numeric criterion is met, but that is not the end of the analysis.
Allowable misclosure = 0.05 × √5.2 = 0.05 × 2.28 = 0.114 ft. The observed misclosure of 0.086 ft is within that limit.
However, the specification limit is a screening test for accidental error. It assumes random error and says nothing about systematic error that happens to be smaller than the limit. Two facts in the record point to systematic effects: a collimation error of 0.011 ft per 100 ft, and a consistent imbalance with backsights about 10 ft longer than foresights. A further complication is that the rod bubble was taped for the last 12 setups, so the rod may have been out of plumb, and rod tilt produces a reading error that grows with the rod reading.
Before accepting the loop, check: the two-peg test history of the instrument, the sum of backsight and foresight distances, any section between benchmarks with a known elevation, and repeat observations at a few setups. A misclosure within tolerance is necessary but not sufficient; it can conceal a systematic error that is nearly as large as the allowance.
2. Estimate the systematic contribution from collimation error and from the unbalanced sights, and prescribe the field correction.
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Quantify the systematic term. Collimation error per setup equals 0.011 ft per 100 ft multiplied by the difference between backsight and foresight distances.
With balances averaging 10 ft and 44 setups, Σ(backsight distance − foresight distance) ≈ 440 ft.
Systematic error ≈ 0.011 × (440 / 100) = 0.011 × 4.40 ≈ 0.048 ft
That is roughly half the observed misclosure, with the sign depending on which sight is longer and the direction of the line-of-sight error. Balanced sights would have cancelled it entirely; this is why balancing is a procedural requirement, not a housekeeping detail.
Field corrections: repair or adjust the instrument and repeat the two-peg test until the error is within the manufacturer's tolerance; replace the cracked rod bubble or use a rod level; record backsight and foresight distances at every setup and balance them to within a few feet; use turning plates and a target on long or steep shots; and reobserve the sections most affected by the imbalance and by the 12 setups with the defective bubble. Do not simply rely on the tolerance to absorb the outcome.
3. If reobservation is impossible before Friday, should the loop be adjusted and submitted, and on what basis?
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Adjustment is arithmetic; it is not a remedy for systematic error. Distributing 0.086 ft over 44 setups, or proportional to distance, produces residually plausible elevations that still inherit the uncorrected collimation and rod-tilt bias. The result may look clean and be wrong.
If the deadline cannot move, the defensible course is:
- Reobserve a subset of setups with a properly adjusted instrument and a sound rod to bound the systematic error. Even a few hours on the worst section converts an unknown bias into a known quantity.
- Adjust the loop using a stated method, and document the method.
- Qualify the deliverable: report the misclosure, the two-peg test results, the sight imbalance, the defective rod bubble, and the bounded systematic error. Do not certify third-order accuracy for the affected sections without evidence.
- Recommend a re-run of the loop, or at least the suspect sections, before construction layout depends on the elevations.
For a drainage design, elevations drive grades and ponding, so the professional risk of submitting a quietly adjusted, systematically biased loop is high. Disclose and let the engineer decide with the actual uncertainty in hand.