Traverse Misclosure in a Village Abadi
A closed traverse ABCDE was run around a village abadi in Muzaffarpur district to fix boundary pillars for a cadastral sheet. The legs were measured with a 30 m steel tape and the angles with a 20-second theodolite. The field computation gives Σ latitude = +0.240 m and Σ departure = -0.180 m. The perimeter is 1,240 m. The sum of absolute latitudes is 3,860 m and the sum of absolute departures is 2,940 m. Leg CD is 210.00 m long, with latitude +180.00 m and departure -95.00 m. The angular closure is within the permissible limit of 40 seconds. The surveyor must submit the closing sheet tomorrow, and the sarpanch wants the plot areas by evening. One field hand suggests shifting the coordinates of the last station to close the figure; another wants to apply the transit rule.
1. Compute the closing error and relative precision. Would you accept the traverse for a 1:1000 cadastral sheet, or require re-observation? State the condition that controls your answer.
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Computation
- Closing error: $e=\sqrt{(+0.240)^2+(-0.180)^2}=\sqrt{0.0576+0.0324}=\sqrt{0.0900}=0.300$ m.
- Relative precision: $P/e=1240/0.300=4133$, i.e. about 1:4,130 (conservatively quote 1:4,000).
- Angular closure is within 40 seconds, so the misclosure is not dominated by an angular blunder; it is consistent with linear random error.
Decision
At 1:1000, 0.300 m plots as 0.30 mm. That is at the edge of a pencil line, so the sheet would look closed. The controlling question is the specification: if the cadastral specification requires 1:5000, 1:4,130 fails and the traverse must be checked or re-observed; if it requires 1:3000, it passes. Do not call it acceptable without naming the standard. In either case, adjust before computing pillar coordinates or areas; a map that looks closed is not proof that the coordinates are good.
2. The field hand proposes the transit rule; the surveyor prefers Bowditch. Which is defensible here? Compute the Bowditch corrections to the latitude and departure of CD.
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Choice
Bowditch (compass) rule is defensible. It assumes the total error is distributed in proportion to leg length, which matches random tape errors accumulating along each leg. The angular closure is already within 40 seconds, so angular error is not dominant, and the traverse is not an extreme case where departures are much larger than latitudes (absolute latitude/departure ratio 3860/2940 = 1.31). The transit rule is an empirical alternative that distributes in proportion to absolute latitude or departure; it is not justified merely because it is available.
Corrections for CD
- Latitude correction: $C_{lat}=-0.240\times(210/1240)=-0.0406$ m.
- Departure correction: $C_{dep}=-(-0.180)\times(210/1240)=+0.0305$ m.
- Corrected latitude = $180.00-0.041=179.959$ m.
- Corrected departure = $-95.00+0.030=-94.970$ m.
Check: the sum of all latitude corrections must be $-0.240$ m and the sum of departure corrections $+0.180$ m, so CD receives its share, not the whole misclosure.
3. Why is shifting the last station to close unacceptable? Draft the closing-sheet note the amin should sign.
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Shifting the last station forces a mathematical closure by inventing a length and bearing for one leg. It concentrates the whole error at E, distorts every area bounded by EA and DE, and destroys the audit trail: the next surveyor cannot tell whether the original measurement or the coordinate was wrong. It also makes a bad traverse look good.
Closing-sheet note (model)
Closed traverse ABCDE, village abadi, Muzaffarpur. Instrument: 20-second theodolite no. ____; tape: 30 m steel no. ____. Perimeter 1,240 m; $\Sigma L=+0.240$ m; $\Sigma D=-0.180$ m; closing error 0.300 m; relative precision 1:4,130; angular closure within 40 seconds. Adjustment by Bowditch rule; leg corrections applied. Areas and pillar coordinates computed from adjusted coordinates. Original field readings retained; no station coordinates forced to close. If specification requires 1:5000, traverse to be re-observed. Date, surveyor, checker.