Closed/open traverse, Bowditch's rule

Closed and open traverses

• Closed traverse: a traverse that ends where it began (a closed loop) or that starts and ends on stations of known position (a closed link/connecting traverse). It gives a mathematical check on the field work.
• Open traverse: a traverse that starts at a known station and ends at a point of unknown position, so there is no closing check. It is used for roads, canals, pipelines and similar route surveys, and the angles and lengths must be checked by repeat measurement.

Bowditch's rule

Bowditch (compass) rule: the closing error in latitude and departure is distributed among the lines in proportion to their lengths.

$$C_L=-\Sigma L_{at}\times\frac{l}{\Sigma l},\qquad C_D=-\Sigma D_{ep}\times\frac{l}{\Sigma l}$$

where $l$ is the length of the line and $\Sigma l$ the perimeter (or total length of the traverse).

Assumptions:

• Angular and linear measurements are of equal precision, and the errors are accidental (random) only.
• The error in the linear measurement of a line is proportional to $\sqrt{l}$, and the error in angular measurement is inversely proportional to $\sqrt{l}$. On these assumptions, the correction to the latitude and departure of each line works out to be proportional to its length.
• The angular misclosure has already been distributed among the angles before the rule is applied.

It is simple, and it is the most commonly used rule in plane surveying.

Example

A four-sided closed traverse has ΣL = 494.58 m, with the computed latitudes and departures below. The errors are $e_N=\Sigma\text{Lat}=+0.26$ m and $e_E=\Sigma\text{Dep}=-0.13$ m, so $e=0.291$ m and the precision is $1:1701$.

Check: the corrections add up to $-e_N$ and $-e_E$, so the corrected latitudes and departures sum to zero. For line AB: $C_{lat}=-(+0.26)\times99.97/494.58=-0.053$.