Error propagation
Level-loop misclosure judged against a C√M tolerance and distributed
A four-leg differential level loop that fails to return to its benchmark by 0.040 ft. Test it against C√M for two orders of accuracy, distribute the misclosure by leg length, and carry the adjusted elevations round.
Given
- A closed differential level loop was run from BM ORCHARD through three turning points and back to BM ORCHARD.
- Leg 1, BM ORCHARD to TP-1: elevation difference +12.442 ft, length 0.62 mile.
- Leg 2, TP-1 to TP-2: elevation difference −5.371 ft, length 0.48 mile.
- Leg 3, TP-2 to TP-3: elevation difference −8.918 ft, length 0.75 mile.
- Leg 4, TP-3 to BM ORCHARD: elevation difference +1.887 ft, length 0.55 mile.
- The published elevation of BM ORCHARD is 512.408 ft.
- The specification for the work is ordinary third-order levelling, allowable misclosure C√M with C = 0.05 ft and M in miles. The client has asked whether the loop would also meet a second-order standard, C = 0.035 ft.
Required
- The misclosure of the loop and the allowable misclosure at both standards.
- The correction to each leg and the adjusted elevation differences.
- Adjusted elevations for the three turning points.
Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.
Worked solution
Misclosure of the loop
A loop that returns to its starting benchmark must have elevation differences summing to zero, because every foot climbed is a foot descended. Whatever the sum actually comes to is the misclosure, and it is the only measure of the loop's quality available from within the loop itself.
Sum the four legs: the loop closes 0.040 ft high, meaning the run around the loop gained 0.040 ft that it should not have. The sign matters for the correction and is easy to lose; write it down explicitly rather than working with magnitudes.
Sum the leg lengths too. Total distance is what the tolerance is written against, and it is also what the corrections will be apportioned by.
misclosure = +12.442 − 5.371 − 8.918 + 1.887 = +0.040 ft
total length M = 0.62 + 0.48 + 0.75 + 0.55 = 2.40 miles
the loop closes 0.040 ft HIGH, so the corrections must be negativeTest against the tolerance
The allowable misclosure for differential levelling is written as C√M. The square-root form is the same series propagation as everywhere else in this subject: levelling error accumulates with the number of instrument setups, the number of setups is proportional to the distance run, and the error of a sum of independent setups grows as the square root of their number. So the allowable grows as the square root of distance, not in proportion to it.
At the third-order coefficient of 0.05 ft, 2.40 miles allows 0.077 ft. The observed 0.040 ft is a little over half of that, so the loop passes comfortably.
Test the second-order standard as well, since it was asked for. At C = 0.035 the allowance is 0.054 ft, and 0.040 ft still passes — though with much less margin. Note that meeting a second-order allowable misclosure is necessary for second-order work but not sufficient: that standard also prescribes sight lengths, balancing, instrument and rod calibration and the number of independent runs, none of which a closure figure can demonstrate.
third order: allowable = 0.05 √2.40 = 0.05 × 1.5492 = 0.077 ft → 0.040 < 0.077 PASS
second order: allowable = 0.035 √2.40 = 0.035 × 1.5492 = 0.054 ft → 0.040 < 0.054 PASS
ratio to the third-order allowable = 0.040 / 0.077 = 0.52Distribute the misclosure by leg length
The correction to each leg is the negative of the misclosure times that leg's share of the total distance. Distribution is by length, not by the size of the elevation difference, and the distinction is real: leg 4 climbs only 1.887 ft but is 0.55 mile long, so it takes a larger correction than its small rise suggests.
The reasoning behind that choice is the same square-root propagation. Error enters at each instrument setup, and the number of setups on a leg is governed by how far the leg runs, not by how much it climbs. A leg twice as long has had twice the opportunity to go wrong.
Round each correction to 0.001 ft, then check the column sums to exactly the negative of the misclosure. If rounding leaves it a thousandth short, put the odd thousandth on the longest leg.
| Leg | Length (mi) | Share of M | Observed Δh (ft) | Correction (ft) | Adjusted Δh (ft) |
|---|---|---|---|---|---|
| BM ORCHARD – TP-1 | 0.62 | 0.2583 | +12.442 | −0.010 | +12.432 |
| TP-1 – TP-2 | 0.48 | 0.2000 | −5.371 | −0.008 | −5.379 |
| TP-2 – TP-3 | 0.75 | 0.3125 | −8.918 | −0.013 | −8.931 |
| TP-3 – BM ORCHARD | 0.55 | 0.2292 | +1.887 | −0.009 | +1.878 |
| Σ | 2.40 | 1.0000 | +0.040 | −0.040 | 0.000 |
Carry the adjusted elevations round the loop
Start at the published benchmark elevation and add each adjusted difference in turn. Because the adjusted differences sum to exactly zero, the final leg must return to 512.408 ft — and if it does not, the arithmetic of the adjustment is wrong, not the field work.
Report the turning-point elevations to 0.001 ft, which is what the notes carried. Note that the largest correction applied anywhere on the loop is 0.013 ft, so no elevation moved by more than about a hundredth of a foot: on a loop that closes this well, adjustment is a formality that makes the elevations mutually consistent rather than a substantive change to any of them.
| Point | Adjusted Δh in (ft) | Elevation (ft) |
|---|---|---|
| BM ORCHARD | 512.408 | |
| TP-1 | +12.432 | 524.840 |
| TP-2 | −5.379 | 519.461 |
| TP-3 | −8.931 | 510.530 |
| BM ORCHARD (return) | +1.878 | 512.408 |
Answer
- Misclosure = +0.040 ft over a loop length of 2.40 miles: the loop returns 0.040 ft high.
- Third-order allowable = 0.05√2.40 = 0.077 ft. The loop passes, at 52 per cent of the allowance.
- Second-order allowable = 0.035√2.40 = 0.054 ft. The loop also passes on misclosure alone, but closure is only one of that standard's requirements.
- Corrections, apportioned by leg length: −0.010, −0.008, −0.013 and −0.009 ft, summing to −0.040 ft.
- Adjusted elevation differences: +12.432, −5.379, −8.931, +1.878 ft, summing to 0.000.
- Adjusted elevations: TP-1 = 524.840 ft, TP-2 = 519.461 ft, TP-3 = 510.530 ft, returning to BM ORCHARD at 512.408 ft.
Check
Run the adjusted differences round the loop in sequence: 512.408 + 12.432 = 524.840; 524.840 − 5.379 = 519.461; 519.461 − 8.931 = 510.530; 510.530 + 1.878 = 512.408. The loop returns to the published benchmark elevation exactly, which it must once the adjusted differences sum to zero.
Check the distribution proportions independently: leg 3 is 0.75 / 2.40 = 0.3125 of the loop, and 0.3125 × 0.040 = 0.0125 ft, which rounds to the 0.013 ft applied. Each other leg checks the same way, and the four shares sum to 1.0000.
Check the corrections have the right sign by asking what they must do: the loop came back 0.040 ft high, so every leg must be reduced. All four corrections are negative. A correction column with mixed signs would mean the misclosure was applied with its own sign rather than its negative.
Check the tolerance formula against a different loop length to make sure it is being applied per-loop rather than per-leg: a 9.60 mile loop, four times as long, would allow 0.05√9.60 = 0.155 ft — twice the allowance for four times the distance, which is the square-root behavior the propagation predicts.
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