Plane and Applied Surveying 2 Traversing Theory Part Traverse Computations  Definition  Types of Meridian  Applications of traversing  Bearings Correction for observed angles (closed traverse) Check angular Misclosure Adjust angular Misclosure Calculate adjusted bearings Compute (E, N) for each traverse line  Coordinates. -Traversing Methods of conducting Traverse 1. Theodolite 2. Total Station 2. Compass 3. GPS Bearings Bearings Bearing is the angle which a certain line make with a certain meridian. Bearing with respect to true meridian is called true bearings while magnetic bearing is the angle which a line makes with respect to magnetic meridian. There are two ways to represent the bearings,  Fore and back bearings  Whole circle bearing (W.C.B) ,(Azimuth)  Reduced Bearing (R.B) or quadrant bearing 6 The bearing of a line measured in the forward direction of survey line is called the ‘Fore Bearing’ (FB) of that line. The bearing of the line measured in the direction opposite to the direction of the progress of survey is called the ‘Back Bearing’ (BB) of the line. BB= FB ± 180° + sign is applied when FB is < 180° - sign is applied when FB is > 180° 1) Whole Circle Bearing (W.C.B) (Azimuth) Is the bearing always measured from north in clockwise direction to a point. Whole Circle Bearing (W.C.B) (Azimuth) 2) Reduced Bearing Reduced bearing or Quadrant bearing is the angle which a line makes from North or South Pole whichever may be near. The value of angle is from 0° to 90° , and are taken either clock wisely or anti clock wisely. -Quadrant bearing The difference between the whole circle bearing and quadrant bearing are as follows. -Example The following fore bearings were observed for lines, AB, BC, CD, and DE Determine their back bearings: • 145°, 285°, 65°, 215° Example The Fore Bearing of the following lines are given Find the Back Bearing. (a) FB of AB= 310° 30’ (b) FB of BC= 145° 15’ (c) FB of CD = 210° 30’ (d) FB of DE = 60° 45’ Example: Convert the following whole circle bearing to quadrant or reduced bearings : ( i ) 42ᵒ 30’ ( ii ) 126ᵒ 15’ ( iii ) 242ᵒ 45’ ( iv ) 328ᵒ10’ Example Convert the following reduced bearings to whole circle bearings: ( I ) N 65ᵒ 12’ E ( ii ) S 36ᵒ 48’ E ( iii ) S 38ᵒ 18’ W ( iv ) N 26ᵒ 32’ W Closed Traverse • Ends at a known point with known direction Geometrical Constraints -Adjust the deflection angles 2-Interior angles Traverse Interior angles are measured clockwise or counterclockwise between two adjacent lines on the inside of a closed polygon figure. Example The following traverse have five sides with five internal angles. Find the angular misclosure and apply the angle correction -3-Exterior angle Traverse Correction for observed angles (closed traverse) Example: IF ∑observed angles for traverse (ABCDA)= 360˚00′ 48″ find misclosure and correct the interior angles. Check Allowable Angle Misclosure Prepared by:Asst. Prof. Salar K.Hussein Erbil Polytechnic University

1. Plane and Applied Surveying 2
Traversing Theory Part
1
Erbil Polytechnic University
Technical Engineering College
Civil Engineering Department
Compass
Prepared by
Assist. Prof. Salar Khudhur Hussein
Assist. Lecturer Mr. Kamal Yaseen

2. Traverse Computations
 Definition
 Types of Meridian
 Applications of traversing
 Bearings
Correction for observed angles (closed traverse)
Check angular Misclosure
Adjust angular Misclosure
Calculate adjusted bearings
Compute (E, N) for each traverse line
 Coordinates.
2

3. Definitions
• Horizontal angle between two points:
• Is the angle between the projection on the horizontal plane of the lines passing through these
points and meeting at third point.
• Meridian: is a reference direction with respect to which the direction of lines are mentioned. OR
• an imaginary line between the North Pole and the South Pole, drawn on maps to help to show
the position of a place:
• There are three types of meridian
• Magnetic Meridian: a direction given by a freely suspended magnetic needle.
• True Meridian: Determined by Astronomical Observation.
• Assumed Meridian: Any direction to be taken as reference line.
• Whole Circle Bearing(WCB):
• Is the bearing always measured from north in clockwise direction to a point.
• Reduced Bearing(RB):
• Is the bearing measured either from North or South in clockwise or counterclockwise direction
whichever pole is near.
3

4. Traversing
Traverse
Series of straight lines connecting survey stations (begins with 2 known
points as baseline)
Determining (E, N) by measuring horizontal angles & distances
 Classification:
Closed and Open Traverse
Closed traverse: When the lines form a circuit that ends at the starting point, it is
known as a closed traverse.
Open traverse: When the lines form a circuit that ends elsewhere except starting
point, it is said to be an open traverse.
4

5. Methods of conducting Traverse
1. Theodolite
2. Total Station
2. Compass
3. GPS
5

6. Bearings
Bearings
Bearing is the angle which a certain line make with a
certain meridian. Bearing with respect to true meridian is
called true bearings while magnetic bearing is the angle
which a line makes with respect to magnetic meridian.
There are two ways to represent the bearings,
 Fore and back bearings
 Whole circle bearing (W.C.B) ,(Azimuth)
 Reduced Bearing (R.B) or quadrant bearing
6

7. The bearing of a line measured in the forward direction of survey line is
called the ‘Fore Bearing’ (FB) of that line.
The bearing of the line measured in the direction opposite to the direction
of the progress of survey is called the ‘Back Bearing’ (BB) of the line.
BB= FB ± 180°
+ sign is applied when FB is < 180°
- sign is applied when FB is > 180°
Fore Bearing and Back Bearing
Fore-Bearing = Back Bearing ± 180°

8. Definitions
To calculate the Whole Circle Bearing (WCB). Azimuth of the
lines from included angles.
Follow one of the following:
1. If the survey direction is clockwise:
WCB of the line = BB of preceding line – included angle.
2. If the survey direction is anti clockwise(counterclockwise)
WCB of the line = BB of preceding line + included angle.
8

9. 1) Whole Circle Bearing (W.C.B) (Azimuth)
Is the bearing always measured from north in clockwise direction to a
point.
9

10. Whole Circle Bearing (W.C.B) (Azimuth)
2) Reduced Bearing
Reduced bearing or Quadrant bearing is the angle which a line
makes from North or South Pole whichever may be near. The value of angle is
from 0° to 90° , and are taken either clock wisely or anti clock wisely.
 The bearing of lines which fall in 1st and 4th Quadrant are measured with
respect to north line is nearer than south line.
 A bearing of lines fall in 2nd and 3rd quadrants are measured from south
line as south is the nearer line.
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11. Quadrant bearing
11
The quadrants are represented are as follows
1st quadrant = N – E
2nd quadrant = S – E
3rd quadrant = S – W
4th quadrant = N – W

12. Conversion of WCB to RB
12

13. The difference between the whole circle bearing and quadrant
bearing are as follows.
Whole Circle Bearing Quadrant Bearing
The horizontal angle which is made by the
survey line, with the magnetic north in a
clockwise direction is known as the Whole
circle bearing.
The horizontal angle which is made by a survey line
with the magnetic North or South whichever is near
the line in the eastward or westward direction is
known as quadrant wearing or reduced bearing.
In the whole circle bearing, the magnetic
North line is considered as the reference
line.
In the quadrantal Bearing, both magnetic North as well
as South lines are considered as a reference line.
13

14. 3
In the whole circle bearing only
clockwise angle is taken from the
reference survey line
In the quadrantal bearing both clockwise, as
well as the anticlockwise angle from the
reference line, is taken
4
The value of the whole circle bearing
ranges from 0° to 360°
The value of the quadrant bearing for reduced
bearing ranges from 0° to 90°
5
The example of a whole circle bearing are
30°,45°,80°,120°,230°, and 320°, etc
Example of quadrant bearing or reduced bearing
are N35°E, S49°E, N65°W, S25°W, etc.
14

15. Example
The following fore bearings were observed for lines, AB,
BC, CD, and DE Determine their back bearings:
• 145° , 285° , 65°, 215°
Solution
The difference between fore bearing and the back bearing of a line must be 180 .
Noting that in WCB angle is from 0° to 360°
• we find Back Bearing = Fore Bearing ±180°
+ 180° is used if θ is less than 180° and
– 180° is used when θ is more than 180°
Hence, BB of AB = 145 + 180 = 325 BB of BC = 65 + 180 = 245
BB of CD = 285 – 180 = 105 BB of DE = 215 – 180 = 35
15

16. Example
The Fore Bearing of the following lines are given Find the
Back Bearing.
(a) FB of AB= 310° 30’
(b) FB of BC= 145° 15’
(c) FB of CD = 210° 30’
(d) FB of DE = 60° 45’
Solution:
(a) BB of AB = 310° 30’ – 180° 0’ = 130° 30’
(b) BB of BC = 145° 15’ + 180° 0’ = 325°15’
(c) BB of CD = 210° 30’ – 180° 0’ = 30° 30’
(d) BB of DE = 60° 45’ + 180° 0’ = 240° 45’
16

17. Example:
Convert the following whole circle bearing to quadrant or
reduced bearings :
( i ) 42ᵒ 30’ ( ii ) 126ᵒ 15’
( iii ) 242ᵒ 45’ ( iv ) 328ᵒ10’
Solution:
( i ) W.C.B. = 42ᵒ30’ ,Quadrant bearing
= N 42ᵒ30’E
( ii ) W.C.B = 126ᵒ15’ , Reduced bearing or R.B.
= 180ᵒ - W.C.B. = 180ᵒ - 126ᵒ15’
= S 53ᵒ45’E
( iii ) W.C.B. = 242ᵒ 45’, R.B. = W.C.B. – 180ᵒ
= 242ᵒ 45’ - 180ᵒ= 62ᵒ 45’
= S 62ᵒ 45’W
( iv ) W.C.B. = 328ᵒ 10’, R.B. = 360ᵒ – W.C.B.
= 360ᵒ - 328ᵒ10’
= N 31ᵒ50’ W
17

18. Example
Convert the following reduced bearings to whole circle
bearings:
( I ) N 65ᵒ 12’ E ( ii ) S 36ᵒ 48’ E
( iii ) S 38ᵒ 18’ W ( iv ) N 26ᵒ 32’ W
Solution
( I ) R.B. = N 65ᵒ 12’ E , W.C.B. = R.B. = 65ᵒ 12’
( ii ) R.B = S 36ᵒ 48’ E , W.C.B. = 180ᵒ - R.B.
= 180ᵒ – 36ᵒ 48’ =143ᵒ 12’
( iii ) R.B. = S 38ᵒ 18’ W , W.C.B. = 180ᵒ + R.B.
= 180ᵒ + 38ᵒ 18’ = 218ᵒ 18’
( iv ) R.B. = N 26ᵒ 32’ W , W.C.B = 360ᵒ - R.B.
= 360ᵒ - 26ᵒ 32’ = 333ᵒ 28’
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19. 19
Closed Traverse
• Ends at a known point with known direction
Geometrical Constraints:
• Interior angles of polygon:
Sum theory = (n – 2)180
• Exterior angles: Sum theory = (n +2)180
• Also: closure on E & N 

20. Adjust the deflection angles:
Angle adjustment =
𝑡𝑜𝑡𝑎𝑙 𝑒𝑟𝑟𝑜𝑟
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑛𝑔𝑙𝑒𝑠
check = ∑ all interior angles- (N-2) x 180 = 0°
360° 02’ 30”- 360° =+00° 02’ 30”
Hence error = 0° 2’ 30” 1’ = 60” :. 2’ x 60 = 120” + 30” = + 150”,,,, 360° 02’ 30”-
360° = 0° 2’ 30”
Angle adjustment =
𝑡𝑜𝑡𝑎𝑙 𝑒𝑟𝑟𝑜𝑟
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑛𝑔𝑙𝑒𝑠
Angle correction = - 150"
5
= -30”
α1=140°0 9’ 30” – 30” = 140° 09’ 00”
α2=70° 20’ 00” – 30” = 70° 19’ 30”
α3=49° 00’ 00” – 30” = 48° 59’ 30”
α4=50° 20’ 00” – 30” = 50° 19’ 30”
α5=50° 13’ 00” – 30” = 50° 12’ 30”
Check : (140° 09’ 00”)+(70° 19’ 30”) +(48° 59’ 30”)+ 50° 19’ 30 +50° 12’ 30” -360°= 0° 00’ 00”
20

21. 2-Interior angles Traverse
Interior angles are measured clockwise or counter-
clockwise between two adjacent lines on the inside of a
closed polygon figure.
21
check = (n-2) x 180 -∑ all interior angles = 0°

22. Example
The following traverse have five sides with five internal
angles. Find the angular misclosure and apply the angle
correction if needed.
22

23. Solution:
The sum of internal angles= 101° 24’ 00” + 149° 13’ 00” + 80°
58’ 30” + 116° 19’ 00” + 92° 04’ 30” = 539º 59’ 00”
whereas the sum should be (n-2)x180º = (5-2)x180º=
540º00’00”
:. The angular misclosure = 540º 00’ 00” - 539º 59’ 00”= +01’
00” or 60”.
Therefore the correction to be applied is 60”/5 = +12” per
angle.
The distribution of error equally to
all angles to adjust out error is shown
in Table:
23

24. 3-Exterior angle Traverse
Exterior angles are measured clockwise or counter-
clockwise between two adjacent lines on the outside
of a closed polygon figure.
Sum Exterior Angles = (n+2) x 180°
Check the angles
24
Check= (n+2) x 180°- ∑all external angles = 0°

25. Step 1
Correction for observed angles (closed traverse)
∑The Theoretical values for internal angles
∑The Theoretical values for external angles
Then find Misclosure, the difference between measured
angles and
Theoretical angles∙
Then Check Allowable Angle Misclosure is required
25
(n-2)*180˚
(n+2)*180˚
Misclosure =∑ measured angles -∑ Theoretical angles

26. Example:
IF ∑observed angles for traverse (ABCDA)= 360˚00′
48″ find misclosure and correct the interior angles:
Solution:
misclosure=∑observed angles -∑ Theoretical angles
Diff =360˚00′ 48″ -360˚00’ 00’’ =0˚00′48″
Correction = 48″⁄4=12″(added -12″ for each angle )
26

27. Check Allowable Angle Misclosure
where:
c is the allowable misclosure in seconds
K is a constant that depends on the level of accuracy specified for the survey
n is the number of angles
According Federal Geodetic Control Subcommittee (FGCS)
recommends:
First order 1.7”
Second-order class I 3”
Second-order class II 4.5”
Third-order class I 10”
Third-order class II 20”
27
c K n


28. Closed-loop Traverse
3
4
2
B
1
5
2
3
4
B2
L
L23
L34
L4B
A
28
• Known (E, N): A & B
Bearing of AB known;
•Measure the distance of LB2, L23,L34, L4B & 1,2,3,4 and 5