Traverse Surveying

Introduction to Traverse Surveying

A traverse is a series of connected lines whose lengths and directions are measured in the field. Traversing is a fundamental method used in surveying to establish control networks, locate boundaries, and map topographic details.

Types of Traverses

  • Closed Traverse: A traverse that originates at a point and returns to the same point, forming a closed polygon. This allows for checking of errors.
  • Open Traverse: A traverse that does not return to its starting point or close on another known point. Error checking is difficult.
Closed Traverse

Bearings & Azimuths

The direction of a line is defined by the horizontal angle it makes with a reference meridian (usually True North or Magnetic North).

Whole-Circle Bearing (Azimuth)

Measured clockwise from the North meridian. Angles range from 0° to 360°.

Drag the point to change the azimuth

Azimuth: 45°

Reduced Bearing (Quadrantal)

Measured from either North or South towards East or West. Angles range from 0° to 90°, always with letters indicating the quadrant (e.g., N 45° E).

Drag the point to change the bearing

Bearing: N 45° E

Departure and Latitude

To calculate coordinates and check for closure, lines are broken down into rectangular components.

  • Latitude (Lat): The orthographic projection of a line onto the reference meridian (North-South axis).
    $$ \text{Latitude} = L \cos(\theta) $$

    Northings are positive (+), Southings are negative (-).

  • Departure (Dep): The orthographic projection of a line onto the East-West axis.
    $$ \text{Departure} = L \sin(\theta) $$

    Eastings are positive (+), Westings are negative (-).

Where \( L \) is the length of the line and \( \theta \) is the bearing or azimuth.

LATITUDE --
DEPARTURE --

Linear Error of Closure (LEC)

In a perfectly measured closed traverse, the sum of all Latitudes should be zero, and the sum of all Departures should be zero. In practice, there is always some error. The gap between the starting point and the calculated final point is the Linear Error of Closure.

Condition for a balanced traverse:
$$ \sum \text{Latitude} = 0 \quad \text{and} \quad \sum \text{Departure} = 0 $$

Calculating the Error

If the sums are not zero, we find the error in latitude (\( e_y \)) and departure (\( e_x \)):

$$ e_x = \sum \text{DEP} $$ $$ e_y = \sum \text{LAT} $$

The magnitude of the error is found using the Pythagorean theorem:

$$ \text{LEC} = \sqrt{(\sum \text{DEP})^2 + (\sum \text{LAT})^2} $$

The direction (bearing) of the error is:

$$ \tan(\theta) = \frac{\sum \text{DEP}}{\sum \text{LAT}} $$

Relative Precision

Expresses the accuracy of the survey as a ratio.

$$ \text{Relative Precision} = \frac{\text{LEC}}{\text{Total Distance}} = \frac{1}{\left(\frac{\text{Total Distance}}{\text{LEC}}\right)} $$

Balancing a Traverse

Once the error of closure is determined, it must be distributed among the traverse lines to close the figure mathematically. There are two common rules:

1. Compass Rule (Bowditch's Method)

Used when angular and linear measurements are assumed to be of equal precision. The correction is proportional to the length of the line.

$$ \text{Correction in Lat}_{line} = -(\sum \text{LAT}) \times \frac{\text{Length of line}}{\text{Total Traverse Perimeter}} $$
$$ \text{Correction in Dep}_{line} = -(\sum \text{DEP}) \times \frac{\text{Length of line}}{\text{Total Traverse Perimeter}} $$

2. Transit Rule

Used when angular measurements are considered more precise than linear measurements. The correction is proportional to the latitude or departure of that line.

$$ \text{Correction in Lat}_{line} = -(\sum \text{LAT}) \times \frac{|\text{Lat of line}|}{\sum |\text{LAT}|} $$
$$ \text{Correction in Dep}_{line} = -(\sum \text{DEP}) \times \frac{|\text{Dep of line}|}{\sum |\text{DEP}|} $$

Note: The denominator uses the sum of the absolute values.

Methods of Obtaining Area of a Closed Traverse

After balancing the traverse, you calculate the adjusted coordinates and find the area enclosed.

Double Meridian Distance (DMD) Method

This method uses the departures to calculate "Double Meridian Distances" for each line.

Rules for finding DMD:
  1. The DMD of the first course is equal to the departure of the course itself.
  2. The DMD of any other course is equal to the DMD of the preceding course, plus the departure of the preceding course, plus the departure of the course itself.
  3. Check: The DMD of the last course is numerically equal to its departure but with the opposite sign.
$$ 2 \times \text{Area} = \sum (\text{DMD} \times \text{Adjusted Latitude}) $$
$$ \text{Area} = \frac{1}{2} \left| \sum (\text{DMD} \times \text{Lat}) \right| $$

Double Parallel Distance (DPD) Method

Similar to DMD, but relies on Latitudes instead of Departures.

Rules for finding DPD:
  1. The DPD of the first course is equal to the latitude of the course itself.
  2. The DPD of any other course is equal to the DPD of the preceding course, plus the latitude of the preceding course, plus the latitude of the course itself.
  3. Check: The DPD of the last course is numerically equal to its latitude but with the opposite sign.
$$ 2 \times \text{Area} = \sum (\text{DPD} \times \text{Adjusted Departure}) $$
$$ \text{Area} = \frac{1}{2} \left| \sum (\text{DPD} \times \text{Dep}) \right| $$