Angular Misclosure on a Canal Traverse
A four-sided closed theodolite traverse is run along a canal head regulator at a site in Rohtas district. The instrument is a 20-second vernier theodolite, and angles are read by the repetition method. The interior angles are A 78°15'20", B 101°42'40", C 96°18'00", and D 83°46'40". The traverse legs are about 120 m, 160 m, 140 m, and 180 m. The surveyor wants to distribute the misclosure equally and proceed, because the drawing is due. The checker says angle C should be re-observed because its backsight was taken on a ranging rod that was later found leaning. The field book has no face-right reading for C, and the instrument was moved before the error was noticed. A second bench mark is available for a check.
1. Compute the angular misclosure and the permissible closure for a 20-second theodolite with four angles. Should the surveyor distribute the misclosure or re-observe?
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Angular closure
Sum $=78°15'20"+101°42'40"+96°18'00"+83°46'40"=360°02'40"$. Misclosure $=+2'40"=+160$ seconds.
Permissible closure for a 20-second instrument with four angles is usually $20\sqrt{4}=40$ seconds. The observed misclosure is four times the permissible value, so the set is rejected. Do not distribute it. A 2'40" misclosure is too large to treat as random; the leaning rod at C is a plausible gross error, and the missing face-right reading means C is not independently checked. Re-observe C and preferably the adjacent angles, with the rod plumb and both faces. Only after the re-observed sum is within 40 seconds may a small residual be distributed equally.
2. If the misclosure were not corrected, what lateral error would 2'40" produce at 180 m, and what would it do to a canal centre-line plotted at 1:2000?
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Effect of the uncorrected misclosure
$2'40"=160$ seconds. In radians: $160/206,265=0.000776$ rad. At 180 m, lateral displacement $=0.000776\times180=0.140$ m (14 cm). If the error is carried through the whole 600 m traverse, the end displacement can reach about $0.000776\times600=0.47$ m.
At 1:2000, 14 cm is 0.07 mm on paper, so the map would look acceptable. For setting out a canal centre-line, 14 cm is significant, and a 0.47 m end displacement is not acceptable. The map check and the setting-out check are different; the stricter one governs. This is why the misclosure must be removed by re-observation, not hidden by plotting.
3. What field procedure would you use for the re-observation, and what records must be kept?
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Re-observation procedure
- Centre carefully with the optical plummet, level the plate, and eliminate parallax.
- Sight the ranging rod at its lowest visible point and keep it vertical with a spirit level or plumb bob.
- Read both faces and use repetition for the horizontal angle; record face-left and face-right separately.
- Avoid very short sights; check the instrument for horizontal-axis and collimation error if the discrepancy persists.
Records
Keep the original set, the reason for rejection, the re-observed values for both faces, the mean, the new closure, the instrument number, observer, date, and checker. Do not erase the bad set; strike it out with initials and a note. If the re-observed closure is within 40 seconds, distribute only that residual, equally or by the chosen rule stated in the field book.