Closed Traverse Closes at 1:2300 Against a 1:5000 Registration Requirement
A consultant is establishing a five-station closed traverse A–B–C–D–E around a 1.8 ha institutional plot in Pune, to be used both for a boundary drawing and later for setting out building columns. Observations were taken with a 5-second total station in one face, one set per angle. The prism supplied has a stated constant of −30 mm, but the operator left the instrument setting at 0 mm when the job was started and did not change it. The measured perimeter is 642.8 m. The sum of the interior angles is 539°58′20″ against a theoretical 540°00′00″. After computing latitudes and departures from the observed distances, the closure is ΔE = +0.19 m and ΔN = −0.20 m. The plot boundary must close better than 1:5000 for registration, the client wants the drawing issued this week, and the same coordinates will be handed to the site team for column layout with a tolerance of ±10 mm.
1. Compute the linear misclosure and precision ratio, judge them against the requirement, and decide whether the drawing can be issued. Comment separately on the angular misclosure.
Read the worked answer
Linear misclosure: √(0.19² + 0.20²) = √(0.0361 + 0.0400) = √0.0761 = 0.276 m.
Precision: 0.276 / 642.8 = 1 in 2,329. The 1:5000 requirement permits 642.8 / 5000 = 0.129 m, so the traverse is about 2.1 times outside the limit. At this precision a boundary point can sit 276 mm from where it should be - far beyond any registration tolerance, and roughly 28 times the ±10 mm column-setting tolerance.
Angular misclosure: 540°00′00″ − 539°58′20″ = 1′40″ = 100″. For a 5-second instrument observing one set, a five-leg traverse should close to around 5″ × √5 ≈ 11″. 100″ is roughly nine times the expected random error, which points to a systematic cause rather than ordinary scatter. The old rule of thumb 1′√n (= 2′14″ here) would let this pass, but that rule belongs to 20-second theodolite practice and should not be applied to a 5-second total station.
Decision: do not issue the drawing as a final boundary document. Two independent indicators - a large angular misclosure and a linear misclosure at twice the tolerance - plus a known instrument-setting error say the network must be re-observed or corrected before any adjustment is accepted. If a drawing must go out this week, mark it provisional and state the precision achieved.
2. The misclosures are inconsistent with the instrument's stated precision. Identify the likely systematic causes, rank the order in which you would check them, and quantify the prism-constant effect.
Read the worked answer
Prism-constant effect: entering 0 mm instead of −30 mm shifts every measured distance by 30 mm along its own direction. Across five legs that is up to 0.15 m of along-leg error; the component surviving into the closure depends on how each leg is oriented relative to the closure vector, but the expected magnitude is of the order of 0.03 × √5 ≈ 0.067 m, with a worst case near 0.15 m. With an allowable misclosure of 0.129 m, this single setting can account for roughly half to all of the observed error. It must be corrected before any judgement is made about the fieldwork.
Check order:
- Prism constant and EDM settings in the job file - quickest, and a known error exists here.
- Instrument and target centering: tribrach and optical plummet, and whether the station marks were disturbed between legs. Miscentering produces exactly this signature - good angles, poor closure.
- Collimation and trunnion-axis check-and-adjust; confirm the instrument was levelled and that both faces agree on a test angle.
- EDM scale factor (ground versus projection) and atmospheric correction - small at 642 m, but not zero if the job is tied to a grid.
- Target height or station-height recording errors - these would show mainly in the vertical, not the horizontal, so they are a low priority here.
After correcting the prism constant, recompute the latitudes and departures. If precision then meets 1:5000, adjust and proceed. If it still does not, re-observe the two legs that contribute most to the closure, and re-check the angular work in two faces.
3. If a full re-observation is not possible before the deadline, explain how you would adjust, which adjustment rule you would choose and why, and what you would state as the limitation of the output.
Read the worked answer
Adjustment procedure:
- Correct all distances for the prism constant and reduce slope distances to horizontal.
- Distribute the 100″ angular misclosure equally over the five angles (20″ each), then recompute bearings.
- Recompute latitudes and departures; the residual closure is now the linear misclosure to be distributed.
- Choose the Bowditch (compass) rule, because both angular and linear errors are present and the acceptance requirement is stated on linear closure. Corrections per leg: ΔE_i = −(e_E/P) × L_i and ΔN_i = −(e_N/P) × L_i, with e_E = +0.19 m, e_N = −0.20 m and P = 642.8 m. For a 160 m leg this gives 160 × 0.19 / 642.8 = 0.047 m in easting and −0.050 m in northing. The transit rule is reserved for cases where angular work is demonstrably much better than linear work, which is not established here - and it would leave the systematic distance error untouched.
- Recompute coordinates and check the polygon closes.
State the limitation explicitly on the drawing and in the report: coordinates are adjusted to close internally but carry an uncorrected systematic component; the achieved precision before adjustment was 1:2,329, which does not satisfy the 1:5000 registration requirement; the boundary cannot be certified until re-observation. Do not release these coordinates for column setting out - an adjustment redistributes error, it does not remove a blunder, and a 30 mm prism error that has been smeared into the coordinates is still there.