LEC13 Traverse Computation and Adjustment

Traverse Computation

Departure and Latitude

• After balancing angles and calculating preliminary azimuths (or bearings), traverse closure is checked by computing the departure and latitude of each line.

• Departure:

  • Orthographic projection on the east-west axis.
  • Equal to the length of the course multiplied by the sine of its azimuth (or bearing) angle.
  • Also called eastings or westings.
  • Formula: $Departure = L \sin \alpha$, where L is the length of the course and $\alpha$ is the azimuth angle.

• Latitude:

  • Orthographic projection on the north-south axis.
  • Equal to the course length multiplied by the cosine of its azimuth (or bearing) angle.
  • Also called northing or southing.
  • Formula: $Latitude = L \cos \alpha$, where L is the length of the course and $\alpha$ is the azimuth angle.

Sign Conventions

• East departures and north latitudes are considered positive (+).
• West departures and south latitudes are considered negative (-).
• When using azimuths (from north) ranging from 0° to 360°, the algebraic signs of sine and cosine functions automatically produce the correct algebraic signs for departures and latitudes.
• When using bearings (angles between 0° and 90°):
  • Sines and cosines are invariably positive.
  • Proper algebraic signs of departures and latitudes must be assigned based on the bearing angle direction.
  • Departure (+), Latitude (+): North-East quadrant
  • Departure (-), Latitude (+): North-West quadrant
  • Departure (+), Latitude (-): South-East quadrant
  • Departure (-), Latitude (-): South-West quadrant

Example

• Calculate the departure and latitude of the following lines:
  • A 10 ft line with an azimuth of 126°55’17”
  • A 20 ft line with an azimuth of 284°35’20”
  • A 10 ft line with a bearing of N 30° E
  • A 10 ft line with a bearing of S 45°20’55” E
  • A 25 ft line with a bearing of S 35° W
  • A 15 ft line with a bearing of S 25° E

Latitude and Departure Misclosure

• Closed-Polygon Traverse:
  • If all angles and distances were measured perfectly, the algebraic sum of the departures of all courses should equal zero.
  • The algebraic sum of all latitudes should equal zero.
• Closed Link-Type Traverses:
  • The algebraic sum of departures ($\Delta X$) should equal the total difference in departure between the starting and ending control points.
  • The same condition applies to latitudes ($\Delta Y$) in a link traverse.
• Misclosures:
  • Due to imperfect observations and errors, the conditions stated above rarely occur.
  • The amounts by which these conditions fail to be met are termed departure misclosure and latitude misclosure.
  • Values are computed by algebraically summing the departures and latitudes and comparing the totals to the required conditions.

Traverse Linear Misclosure and Relative Precision

• The magnitudes of the departure and latitude misclosures for closed-polygon-type traverses give an indication of the precision that exists in the observed angles and distances.
• Large misclosures indicate significant errors or mistakes.
• Small misclosures usually mean the observed data are precise and free of mistakes, but it is not a guarantee that systematic or compensating errors do not exist.
• Linear Misclosure:
  • Formula: $\sqrt{(departure\,misclosure)^2 + (latitude\,misclosure)^2}$
• Relative Precision:
  • Formula: $\frac{linear\,misclosure}{traverse\,length\,or\,traverse\,perimeter}$

Example: Latitude, Departure, Linear Misclosure, and Relative Position

• Based on preliminary azimuths and lengths, calculate the departures and latitudes, linear misclosure, and relative precision of the traverse.

Traverse Adjustment (Compass or Bowditch Rule)

• The compass, or Bowditch, rule adjusts the departures and latitudes of traverse courses in proportion to their lengths.
• Correction in departure for a course:
  • Formula: $- \frac{total\,departure\,misclosure}{traverse\,perimeter} \times length\,of\,course$
• Correction in latitude for a course:
  • Formula: $- \frac{total\,latitude\,misclosure}{traverse\,perimeter} \times length\,of\,course$
• Balanced Departures and Latitudes:
  • If the calculation results in a negative sign, the length of traverse (latitude or departure) should be decreased: Initial length – correction.
  • If the calculation results in a positive sign, the length of traverse (latitude or departure) should be added: Initial length + correction.
• If the coordinates of one point are known, after balancing departures and latitudes, the coordinates of other points of the traverse can be found:
  • $X<em>B = X</em>A + departure_{AB}$
  • $Y<em>B = Y</em>A + latitude_{AB}$

Example: Balancing Departures and Latitude Using Compass Rule

• Using preliminary azimuths and lengths, compute departures and latitudes, linear misclosure, and relative precision. Balance the departures and latitudes using the compass rule.

References

• Charles D. Ghilani and Paul R. Wolf, Elementary Surveying: An Introduction to Geomatics, 16th edition, Pearson, 2015.