Balancing linked traverse, transit rule

Balancing angular misclosure in a link traverse

A link traverse starts and ends on stations of known coordinates, and the bearings of the first and last reference lines are known. The angular misclosure is therefore found by carrying the bearing through the traverse.

1. Starting from the known bearing of the first reference line, compute the bearing of every line from the observed angles. For clockwise angles, bearing of a line $=$ bearing of previous line $+180^\circ+$ angle; for anticlockwise angles, bearing of previous line $+180^\circ-$ angle (adding or subtracting $360^\circ$ as needed).
2. The computed bearing of the last reference line is compared with its known bearing:

$$\text{Angular misclosure}=\text{computed bearing}-\text{known bearing}$$

3. If it is within the allowable limit (for example $\pm k\sqrt{n}$, $n$ = number of angles), distribute it equally, the correction per angle being $-\text{misclosure}/n$. The first bearing receives one correction, the second two, and so on, so the last bearing receives $n$ times the correction and then equals the known value.

After the angles are balanced, the lengths are balanced through the linear misclosure of coordinates.

Transit rule (balancing latitudes and departures)

Transit rule: the total error in latitude (or departure) is distributed in proportion to the latitude (or departure) of each line, taken without sign.

$$C_L=-\Sigma L_{at}\times\frac{|\text{lat of line}|}{\Sigma|\text{lat}|},\qquad C_D=-\Sigma D_{ep}\times\frac{|\text{dep of line}|}{\Sigma|\text{dep}|}$$

It is used when angles are measured more precisely than lengths (for example with a precise theodolite and ordinary taping). The assumption is that the error in a line's latitude depends on the latitude itself, so a line running mainly north-south gets a large latitude correction and a small departure correction. It is used less often than Bowditch's rule because its corrections depend on the directions of the lines relative to the reference axes, so the result changes if the axes are rotated.