Predicted traverse closure error against the observed misclosure

1Per-course propagation

Each course contributes error to the closure through both its distance and its azimuth. The propagation into latitude and departure is the same as for a single course: partial derivative times standard error, combined in quadrature.

Convert the angular standard error to radians first: 10″ divided by 206,264.81 gives 4.84814 × 10⁻⁵ rad. Multiplied by a course length this is the across-course displacement, and it is the term that grows with the size of the traverse.

On these courses the two sources contribute comparably. The distance term is a flat 0.020 ft, and the angular term D·σ_α runs from 0.018 to 0.025 ft — so neither the distances nor the angles dominate, which is what makes the compass rule the right adjustment for this traverse.

2Course by course

Work the two propagations for each of the five courses and square the results, because it is the variances that will be summed.

Note that σ_lat and σ_dep differ course by course according to the azimuth: on a course running nearly north the angular error lands almost entirely in the departure, and on a course running nearly east it lands almost entirely in the latitude. Across five courses spread around a closed figure those preferences largely even out, which is why the two summed variances come out close to one another.

3Predicted misclosure

The closure in latitude is the sum of five independent latitudes, so its variance is the sum of their variances; likewise for departure. The linear misclosure is the resultant of the two, so its variance is the sum of both.

That gives a predicted standard misclosure of 0.066 ft, and a predicted precision of about 1:33,800. This is what the instrument and the geometry can deliver — not a target, but an expectation.

Understand what the prediction is and is not. It is a root-mean-square figure for a two-dimensional quantity, so a substantial share of otherwise blameless traverses will close worse than it. The value at which a traverse becomes suspect is 1.73 times it — the 95 per cent point of the two-dimensional misclosure distribution — which is 0.114 ft here.

4The whole sum collapses

There is a simplification worth knowing, because it turns a twenty-line table into one line. Add σ_lat² and σ_dep² for a single course and the trigonometry cancels: the distance terms give cos²α + sin²α = 1 times σ_D², and the angular terms give sin²α + cos²α = 1 times (D σ_α)².

So the predicted misclosure of any traverse is the square root of the sum, over all courses, of σ_D² + (D σ_α)². The azimuths drop out entirely. Only the lengths and the two standard errors matter.

Use the long table when a per-course breakdown is wanted — to see which course is the weak one — and the collapsed form when only the total is needed. They must agree exactly, which makes each a check on the other.

5Compare with the field

The traverse actually closed with ΣLat = +0.0488 ft and ΣDep = −0.0467 ft, giving an observed misclosure of 0.0676 ft and a precision of 1:33,074.

Observed over predicted is 1.02. The traverse closed almost exactly as well as the instrument and the geometry say it should — the strongest possible evidence that the misclosure is accumulated random error and contains no blunder. There is nothing to look for and the compass rule may be applied with confidence.

Read the comparison the other way to see its value. Had the same traverse closed at 0.30 ft — a precision of 1:7,400, which many specifications would still pass — that would be 4.5 times the predicted misclosure, which accumulated random error alone would essentially never produce. A specification would have accepted it; the prediction would have caught it. This is the discipline the comparison is for: a tolerance tells you whether a traverse is good enough, and only a prediction tells you whether it is behaving.