Traverse Computations

Traverse Computations

A Traverse is a series of consecutive lines whose lengths and directions are determined from field measurements.

Types of Traverses

1. Open Traverse: Does not return to the starting point or close upon a point of known position. Used for route surveys.
2. Closed Traverse: Begins and ends at the same point (Loop Traverse) or begins and ends at points of known position (Link Traverse). Used for property boundaries.

Latitudes and Departures

• Latitude ($L$): The projection of a line on the North-South meridian. $L = D \cos \alpha$ Where $D$ is distance and $\alpha$ is bearing angle.

  • North Latitude ($+$)
  • South Latitude ($-$)

• Departure ($D_{ep}$): The projection of a line on the East-West line. $D_{ep} = D \sin \alpha$

  • East Departure ($+$)
  • West Departure ($-$)

Error of Closure

In a closed loop traverse, the algebraic sum of latitudes ($\Sigma L$) and departures ($\Sigma D_{ep}$) should be zero. If not, there is an error.

• Linear Error of Closure ($LEC$): $LEC = \sqrt{(\Sigma L)^2 + (\Sigma D_{ep})^2}$

• Relative Error of Closure ($REC$): $REC = \frac{LEC}{\Sigma D}$ Where $\Sigma D$ is the total length (perimeter) of the traverse. Expressed as a ratio (e.g., 1:5000).

Balancing a Traverse

Adjusting the latitudes and departures so their algebraic sums become zero.

1. Compass Rule (Bowditch Rule)

Assumes errors in distance and angle are equal. The correction applied to the latitude or departure of any course is proportional to its length.

$c_L = -\Sigma L \left(\frac{d}{\Sigma D}\right)$ $c_D = -\Sigma D_{ep} \left(\frac{d}{\Sigma D}\right)$

Where:

• $c_L, c_D$: Corrections to latitude and departure.
• $\Sigma L, \Sigma D_{ep}$: Total error in latitude and departure.
• $d$: Length of the specific course.
• $\Sigma D$: Total length of the traverse.

2. Transit Rule

Assumes angular errors are less than linear errors. Corrections are proportional to the latitude or departure itself.

$c_L = -\Sigma L \left(\frac{|L|}{\Sigma |L|}\right)$ $c_D = -\Sigma D_{ep} \left(\frac{|D_{ep}|}{\Sigma |D_{ep}|}\right)$

Solved Problems