CogoKit

Levelling loops

Closed level loop tested against a tolerance and adjusted

A four-leg level loop returns to its starting benchmark 0.060 ft out. Test the misclosure against a C times root-M tolerance, then distribute it in proportion to leg length and compute adjusted elevations.

Apply· about 20 minutes by hand· 6 steps

Given

  • A closed level loop that starts and ends on BM-101, whose published elevation is 1,042.180 ft (US survey feet). Elevations are carried to 0.001 ft.
  • Leg 1, BM-101 to TP-A: observed difference +18.462 ft over 0.80 mile.
  • Leg 2, TP-A to BM-102: observed difference -7.235 ft over 1.20 miles.
  • Leg 3, BM-102 to TP-B: observed difference +11.908 ft over 1.60 miles.
  • Leg 4, TP-B back to BM-101: observed difference -23.075 ft over 1.40 miles.
  • The specification for this order of work allows a misclosure of C times the square root of the loop length in miles, with C = 0.05 ft.

Required

  • The loop misclosure.
  • The allowable misclosure, and whether the work is acceptable.
  • The correction to each leg and the adjusted elevations of TP-A, BM-102 and TP-B.

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 begins and ends on the same monument must show a net elevation difference of zero, because the monument cannot be at two elevations at once. Whatever the observed differences actually sum to is the misclosure, and it is the accumulated effect of every random error in the loop.

Sum the four observed differences with their signs. Do not take absolute values; the signs are what makes the test meaningful.

misclosure = +18.462 - 7.235 + 11.908 - 23.075 misclosure = +0.060 ft total loop length = 0.80 + 1.20 + 1.60 + 1.40 = 5.00 miles

Test against the tolerance

Levelling error accumulates as a random walk, so the expected misclosure grows with the square root of the distance run rather than with the distance itself. That is the whole content of the C times root-M form: doubling the length of a loop should roughly multiply the expected misclosure by 1.414, not by 2.

The coefficient C is set by the order of accuracy being worked to, and it differs between specifications and between jurisdictions, so it must be taken from the job's own specification rather than assumed. Here C = 0.05 ft with the length in miles.

allowable = 0.05 x sqrt(5.00) allowable = 0.05 x 2.2361 = 0.1118 ft |misclosure| = 0.060 ft < 0.1118 ft accept

Distribute the misclosure by leg length

The misclosure is distributed in proportion to leg length, not in proportion to the size of the elevation difference. The reason is physical: the error in a level line grows with the number of instrument setups, and the number of setups tracks the distance run. A short leg with a large elevation change is not the suspect; a long leg is, however gentle its grade.

The correction to each leg is the negative of the misclosure times that leg's share of the total length. Corrections are computed to 0.001 ft and must sum back to the negative of the misclosure.

correction(i) = -misclosure x (length(i) / total length) leg 1: -0.060 x (0.80 / 5.00) = -0.0096 -> -0.010 leg 2: -0.060 x (1.20 / 5.00) = -0.0144 -> -0.014 leg 3: -0.060 x (1.60 / 5.00) = -0.0192 -> -0.019 leg 4: -0.060 x (1.40 / 5.00) = -0.0168 -> -0.017 sum of corrections = -0.060 ft

Adjustment sheet

Applying each correction to its observed difference gives adjusted differences that sum to zero, which is the definition of an adjusted loop.

Level loop adjustment, corrections in proportion to leg length
LegLength (mi)Observed dH (ft)Correction (ft)Adjusted dH (ft)
BM-101 to TP-A0.80+18.462-0.010+18.452
TP-A to BM-1021.20-7.235-0.014-7.249
BM-102 to TP-B1.60+11.908-0.019+11.889
TP-B to BM-1011.40-23.075-0.017-23.092
Totals5.00+0.060-0.0600.000

Carry the adjusted elevations

Starting from the held elevation of BM-101 and applying the adjusted differences in order gives the elevation of each point on the loop. The last step must return exactly to the held value; if it does not, the corrections were mis-signed or mis-added.

TP-A = 1042.180 + 18.452 = 1060.632 BM-102 = 1060.632 - 7.249 = 1053.383 TP-B = 1053.383 + 11.889 = 1065.272 BM-101 = 1065.272 - 23.092 = 1042.180 (closes on the held value)

Reading the result

A misclosure of 0.060 ft in 5.00 miles is a little over half the allowance, which is comfortable but not outstanding. If a job repeatedly lands at 50 to 80 percent of allowance, the usual causes are unbalanced sights, a rod bubble that is out, or turning points that are not firm.

One caution about the adjusted elevations: they are the best estimate available from this loop alone, but the loop is a single closed figure with one redundancy. A benchmark whose elevation matters should be tied by a second, independent route so that the discrepancy between routes can be examined rather than merely smoothed away.

Answer

  • Misclosure = +0.060 ft over 5.00 miles
  • Allowable = 0.05 x sqrt(5.00) = 0.112 ft; the loop is within tolerance
  • TP-A = 1,060.632 ft
  • BM-102 = 1,053.383 ft
  • TP-B = 1,065.272 ft

Check

The four adjusted differences sum to +18.452 - 7.249 + 11.889 - 23.092 = 0.000 ft, so the adjusted loop closes exactly.

Carrying the adjusted differences forward from BM-101 returns 1,042.180 ft, the held elevation, with no residual.

Ratio check on the corrections: leg 3 is twice the length of leg 1 and received -0.019 ft against -0.010 ft, which is the 2:1 proportion expected to within the 0.001 ft rounding.

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