Introduction to Surveying

BASICS OF TRAVERSING

Dr Philip CollierDepartment of GeomaticsThe University of Melbournep.collier@unimelb.edu.auRoom D316

Overview

• In this lecture we will cover : Rectangular and polar coordinates

Definition of a traverse

Applications of traversing

Equipment and field procedures

Reduction and adjustment of data

Rectangular coordinates

E=EB-EA

EA

NA

(EA,NA)

EB

NB

(EB,NB)

Point A

Point B

North

East

N=NB-NA

Polar coordinates

d

North

East

Point A

Point B

~ whole-circle bearingd ~ distance

Whole circle bearingsBearing are measuredclockwise from NORTHand must lie in the range0o 360o

North0o

East90o

South180o

West270o

1st quadrant

2nd quadrant3rd quadrant

4th quadrant

Coordinate conversions

22

1

NEd

NE

tan

cosdN

sindE

Rectangular to polar Polar to rectangular

d

E

N d

E

N

What is a traverse?

• A polygon of 2D (or 3D) vectors

• Sides are expressed as either polar coordinates (,d) or as rectangular coordinate differences (E,N)

• A traverse must either close on itself

• Or be measured between points with known rectangular coordinates

A closedtraverse A traverse between

known points

Applications of traversing

• Establishing coordinates for new points

(,d)

(,d)

(,d)

(E,N)new

(E,N)new

(E,N)known

(E,N)known

Applications of traversing

• These new points can then be used as a framework for mapping existing features

(,d

)

(,d)

(,d)

(,d) (,d)

(,d)

(E,N)new(E,N)new

(E,N)new

(E,N)new

(E,N)new

(E,N)known

(E,N)known

Applications of traversing

• They can also be used as a basis for setting out new work

(E,N)new

(E,N)new

(E,N)known

(E,N)known

Equipment

• Traversing requires : An instrument to measure angles

(theodolite) or bearings (magnetic compass) An instrument to measure distances (EDM

or tape)

Measurement sequence

77.1

999.92

60.6

3

129.76

32.20

A

B

C

D

E

205

o

21

o

232o

56o

352

o

168

o

48o

232o

303 o

118o

Computation sequence1. Calculate angular misclose

2. Adjust angular misclose

3. Calculate adjusted bearings

4. Reduce distances for slope etc…

5. Compute (E, N) for each traverse line

6. Calculate linear misclose

7. Calculate accuracy

8. Adjust linear misclose

Calculate internal angles

PointForesigh

tBearing

Backsight

Bearing

InternalAngle

AdjustedAngle

A 21o 118o 97o

B 56o 205o 149o

C 168o 232o 64o

D 232o 352o 120o

E 303o 48o 105o

=(n-2)*180

Misclose

Adjustment

At each point :• Measure foresight bearing• Meaure backsight bearing• Calculate internal angle (back-

fore)

For example, at B :• Bearing to C = 56o

• Bearing to A = 205o

• Angle at B = 205o - 56o = 149o

Calculate angular misclose

PointForesigh

tBearing

Backsight

Bearing

InternalAngle

AdjustedAngle

A 21o 118o 97o

B 56o 205o 149o

C 168o 232o 64o

D 232o 352o 120o

E 303o 48o 105o

=(n-2)*180

535o

Misclose -5o

Adjustment -1o

Calculate adjusted angles

PointForesigh

tBearing

Backsight

Bearing

InternalAngle

AdjustedAngle

A 21o 118o 97o 98o

B 56o 205o 149o 150o

C 168o 232o 64o 65o

D 232o 352o 120o 121o

E 303o 48o 105o 106o

=(n-2)*180

535o 540o

Misclose -5o

Adjustment -1o

Compute adjusted bearings

• Adopt a starting bearing• Then, working clockwise around the traverse :

Calculate reverse bearing to backsight (forward bearing 180o)

Subtract (clockwise) internal adjusted angle Gives bearing of foresight

• For example (bearing of line BC) Adopt bearing of AB 23o

Reverse bearing BA (=23o+180o) 203o

Internal adjusted angle at B 150o

Forward bearing BC (=203o-150o) 53o

Compute adjusted bearings

Line

Forward Bearing

Reverse Bearing

Internal Angle

AB 23o 203o 150o

BC 53o

CD

DE

EA

AB

203

o

53o

A

B

C

D

E

150o

Compute adjusted bearings

Line

Forward Bearing

Reverse Bearing

Internal Angle

AB 23o 203o 150o

BC 53o 233o 65o

CD 168o

DE

EA

AB

23

o

233o

168

oA

B

C

D

E

65o

Compute adjusted bearings

Line

Forward Bearing

Reverse Bearing

Internal Angle

AB 23o 203o 150o

BC 53o 233o 65o

CD 168o 348o 121o

DE 227o

EA

AB

23

o

53o 3

48

o

227o

A

B

C

D

E

121o

Compute adjusted bearings

Line

Forward Bearing

Reverse Bearing

Internal Angle

AB 23o 203o 150o

BC 53o 233o 65o

CD 168o 348o 121o

DE 227o 47o 106o

EA-59o

301o

AB

23

o

53o 1

68

o

47o

301 o

A

B

C

D

E

106o

Compute adjusted bearings

Line

Forward Bearing

Reverse Bearing

Internal Angle

AB 23o 203o 150o

BC 53o 233o 65o

CD 168o 348o 121o

DE 227o 47o 106o

EA 301o 121o 98o

AB 23o (check)

23

o

53o 1

68

o

227o

121 o

A

B

C

D

E

98o

(E,N) for each line

• The rectangular components for each line are computed from the polar coordinates (,d)

• Note that these formulae apply regardless of the quadrant so long as whole circle bearings are used

cosdN

sindE

Vector components

Line Bearing Distance

E N

AB 23o 77.19 30.16 71.05

BC 53o 99.92 79.80 60.13

CD 168o 60.63 12.61 -59.31

DE 227o 129.76 -94.90 -88.50

EA 301o 32.20 -27.60 16.58

(399.70) (0.07) (-0.05)

Linear misclose & accuracy

• Convert the rectangular misclose components to polar coordinates

• Accuracy is given by

22

1

NEd

NE

tan

)miscloselinear/lengthtraverse(:1

Beware of quadrant whencalculating using tan-1

N

E

positiveadd 180o

Quadrants and tan function

+

negativeadd 180o

+

negativeadd 360o

+

+

positiveokay

For the example…

• Misclose (E, N) (0.07, -0.05)

• Convert to polar (,d) = -54.46o (2nd quadrant) = 125.53o d = 0.09 m

• Accuracy 1:(399.70 / 0.09) = 1:4441

Bowditch adjustment

• The adjustment to the easting component of any traverse side is given by :

Eadj = Emisc * side length/total perimeter

• The adjustment to the northing component of any traverse side is given by :

Nadj = Nmisc * side length/total perimeter

The example…

• East misclose 0.07 m• North misclose –0.05 m• Side AB 77.19 m• Side BC 99.92 m• Side CD 60.63 m• Side DE 129.76 m• Side EA 32.20 m• Total perimeter 399.70 m

Vector components (pre-adjustment)

Side

E N dE dN Eadj Nadj

1A 30.16 71.05

AB 79.80 60.13

BC 12.61 -59.31

CD -94.90 -88.50

D1 -27.60 16.58

Misc

(0.07) (-0.05)

The adjustment componentsSide

E N dE dN Eadj Nadj

1A 30.16 71.05 0.014 -0.010

AB 79.80 60.13 0.016 -0.012

BC 12.61 -59.31 0.011 -0.008

CD -94.90 -88.50 0.023 -0.016

D1 -27.60 16.58 0.006 -0.004

Misc

(0.07) (-0.05)(0.070)

(-0.050)

Adjusted vector componentsSide

E N dE dN Eadj Nadj

1A 30.16 71.05 0.014 -0.010 30.146 71.060

AB 79.80 60.13 0.016 -0.012 79.784 60.142

BC 12.61 -59.31 0.011 -0.008 12.599 -59.302

CD -94.90 -88.50 0.023 -0.016 -94.923 -88.484

D1 -27.60 16.58 0.006 -0.004 -27.606 16.584

Misc

(0.07) (-0.05) 0.070 -0.050 (0.000)(0.000)

Introduction to Surveying

BASICS OF TRAVERSING

Dr Philip CollierDepartment of GeomaticsThe University of Melbournep.collier@unimelb.edu.auRoom D316