7.3 Traverse Closure, Error, and Relative Precision
Key Takeaways
- A closed traverse should have algebraic sums of latitude and departure equal to zero after perfect observations.
- Linear misclosure is the square root of latitude misclosure squared plus departure misclosure squared.
- Relative precision is commonly expressed as total traverse length divided by linear misclosure.
- Closure checks do not prove the survey is correct; they only evaluate internal consistency of the measured traverse.
Traverse Closure and Precision Checks
A closed traverse is expected to return to its starting point or to another known control point. In a perfect world, the sum of all latitudes would be zero and the sum of all departures would be zero. Field work is not perfect, so the computed endpoint usually misses the known endpoint by a small amount. The FS exam often asks you to compute that miss and interpret it without overclaiming what it proves.
Latitude misclosure is the algebraic sum of all north-south components. Departure misclosure is the algebraic sum of all east-west components. If a traverse starts and ends at the same control point, those sums should be zero before error. If it starts at one known point and ends at another, compare the computed coordinate difference to the known coordinate difference. The principle is the same: observed coordinate change minus known coordinate change equals misclosure.
| Quantity | Computation | Meaning |
|---|---|---|
| Sum latitude | Add signed latitudes | North-south computed change |
| Sum departure | Add signed departures | East-west computed change |
| Latitude misclosure | Computed latitude change minus known change | North-south residual |
| Departure misclosure | Computed departure change minus known change | East-west residual |
| Linear error of closure | Square root of lat error squared plus dep error squared | Magnitude of miss |
| Relative precision | Total traverse length divided by linear closure error | Ratio used to compare traverse quality |
For a closed traverse with latitude sum +0.22 ft and departure sum -0.31 ft, the linear error of closure is sqrt(0.22 squared + -0.31 squared), or about 0.38 ft. If the total traverse length is 3850 ft, the relative precision is 3850/0.38, about 1:10100. The ratio is usually reported as one part in the rounded denominator. A larger denominator means a smaller closure error relative to the work length.
Closure can be misleading. A traverse may close numerically even if it was connected to the wrong monument, used the wrong basis of bearing, or had compensating errors. A poor angular observation can be hidden by a distance error in the opposite direction. On the exam, do not choose an answer that treats good closure as proof of boundary correctness or legal sufficiency. Closure is an internal mathematical check.
Angular closure is related but separate. For a polygon with n sides, the theoretical sum of interior angles is (n - 2) times 180 deg. If the measured sum differs, the angular misclosure can be distributed before computing directions, depending on the adjustment method and problem instructions. If a problem gives final bearings or azimuths, you may not need to adjust angles at all.
A practical FS workflow is:
- Convert all directions to azimuths.
- Compute each course latitude and departure with signs.
- Sum the signed latitudes and departures.
- Compare to the expected coordinate change.
- Compute linear error of closure.
- Divide total traverse length by closure error for relative precision.
- Adjust only if the problem asks for an adjusted traverse.
Watch rounding. If a table is rounded to 0.01 ft, the closure from the table may differ slightly from closure using full calculator precision. Use the precision implied by the choices. When answers are close, carry extra digits through latitudes and departures, then round at the end. In production, this is also why spreadsheets store full precision while reports show a reasonable number of decimals.
A closed traverse has latitude misclosure +0.30 ft and departure misclosure -0.40 ft. What is the linear error of closure?
A traverse length is 6000 ft and its linear error of closure is 0.50 ft. What is the relative precision?
What does a good numerical traverse closure prove?