Closed traverse: latitudes, departures and Bowditch's rule

A traverse is a chain of survey lines whose lengths and directions are measured. In a closed traverse the chain returns to its start or to another known point, so its errors can be found and adjusted.

Latitude and departure

Each line of length L with whole circle bearing θ is resolved into north–south and east–west components:

Latitude = L cos θ Departure = L sin θ

North latitudes and east departures are positive.

Closing error

For a perfect closed traverse, ΣL = 0 and ΣD = 0. In reality they don't quite:

e = √( (ΣL)² + (ΣD)² )

Precision is expressed as 1 in (perimeter / e). Typical targets are 1 in 3000 for ordinary work and 1 in 10000 or better for control.

Bowditch's (compass) rule

Used when angles and distances are measured with similar precision, as with a total station or compass and tape:

Correction to latitude of a line = − ΣL × length of lineperimeter

Correction to departure of a line = − ΣD × length of lineperimeter

Transit rule

When angles are measured more precisely than distances (theodolite and tape), corrections are made in proportion to each latitude or departure instead:

Correction to latitude = − ΣL × latitude of the line, ignoring signsum of all latitudes, ignoring sign

Worked example

A traverse of perimeter 1200 m has ΣL = +0.36 m and ΣD = −0.48 m.

e = √(0.36² + 0.48²) = 0.60 m Precision = 1 in 12000.60 = 1 in 2000

A 300 m line gets a latitude correction of −0.36 × 300/1200 = −0.09 m and a departure correction of +0.48 × 300/1200 = +0.12 m.

Coordinates

Once adjusted, add latitudes and departures cumulatively from a known point to get the northing and easting of every station. Areas follow from the coordinates.