Transcript 451-102 Introduction to Surveying (BPD)

H.U. MINING ENGINEERING DEPARTMENT
MAD 256 – SURVEYING
BASICS OF TRAVERSING
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 closed
traverse
A traverse between
known points
Applications of traversing
• Establishing coordinates for new points
(E,N)known
(E,N)known
(E,N)new
(E,N)new
Applications of traversing
• These new points can then be used as a
framework for mapping existing features
(E,N)known
(E,N)new
(E,N)new
(E,N)new
(E,N)new
(E,N)new
(E,N)known
Applications of traversing
• They can also be used as a basis for setting
out new work
(E,N)known
(E,N)known
(E,N)new
(E,N)new
Equipment
• Traversing requires :


An instrument to measure angles (theodolite) or
bearings (magnetic compass)
An instrument to measure distances (EDM or tape)
Measurement sequence
C
B
D
A
E
Computation sequence
1. 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
Point
Foresight Backsight
Azimuth
Azimuth
Internal
Angle
A
21o
118o
97o
B
56o
205o
149o
C
168o
D
232o
E
303o
 =(n-2)*180
Misclose
Adjustment
Adjusted
Angle
o
o
232
64
At each point :
• oMeasure foresight
azimuth
352
120o
• Meaure backsight azimuth
o
• o Calculate 105
internal
angle (back-fore)
48
For example, at B :
• Azimuth to C = 56o
• Azimuth to A = 205o
• Angle at B = 205o - 56o = 149o
Calculate angular misclose
Point
Foresight Backsight
Azimuth
Azimuth
Internal
Angle
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
Adjusted
Angle
Calculate adjusted angles
Point
Foresight Backsight
Azimuth
Azimuth
Internal
Angle
Adjusted
Angle
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
535o
540o
 =(n-2)*180
Misclose
-5o
Adjustment
-1o
Compute adjusted azimuths
• Adopt a starting azimuth
• Then, working clockwise around the traverse :



Calculate reverse azimuth to backsight (forward azimuth 180o)
Subtract (clockwise) internal adjusted angle
Gives azimuth of foresight
• For example (azimuth of line BC)




Adopt azimuth of AB
Reverse azimuth BA (=23o+180o)
Internal adjusted angle at B
Forward azimuth BC (=203o-150o)
23o
203o
150o
53o
Compute adjusted azimuths
C
B
150
o
D
Line
Forward
Azimuth
Reverse
Azimuth
Internal
Angle
AB
23o
203o
150o
BC
53o
CD
DE
EA
A
E
AB
Compute adjusted azimuths
C
65o
B
D
Line
Forward
Azimuth
Reverse
Azimuth
Internal
Angle
AB
23o
203o
150o
BC
53o
233o
65o
CD
168o
DE
EA
A
E
AB
Compute adjusted azimuths
C
B
121o
D
Line
Forward
Azimuth
Reverse
Azimuth
Internal
Angle
AB
23o
203o
150o
BC
53o
233o
65o
CD
168o
348o
121o
DE
227o
EA
A
E
AB
Compute adjusted azimuths
C
B
D
A
106o
E
Line
Forward
Azimuth
Reverse
Azimuth
Internal
Angle
AB
23o
203o
150o
BC
53o
233o
65o
CD
168o
348o
121o
DE
227o
47o
106o
EA
AB
-59o
301o
Compute adjusted azimuths
C
B
D
A
98o
E
Line
Forward
Azimuth
Reverse
Azimuth
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)
(E,N) for each line
• The rectangular components for each line are
computed from the polar coordinates (,d)
E  d sin 
N  d cos 
• Note that these formulae apply regardless of
the quadrant so long as whole circle bearings
are used
Vector components
Line
Azimuth
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
1  E 
  tan 

 N 
2
2
d  E  N
Beware of quadrant when
calculating  using tan-1
• Accuracy is given by
1 : (traverse length / linear misclose)
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
North misclose
Side AB
Side BC
Side CD
Side DE
Side EA
Total perimeter
0.07
–0.05
77.19
99.92
60.63
129.76
32.20
399.70
m
m
m
m
m
m
m
m
Vector components
Side
E
N
AB
30.16
71.05
BC
79.80
60.13
CD
12.61
-59.31
DE
-94.90
-88.50
EA
-27.60
16.58
Misc
(0.07)
(-0.05)
dE
(pre-adjustment)
dN
Eadj
Nadj
The adjustment components
Side
E
N
dE
dN
AB
30.16
71.05
0.014
-0.010
BC
79.80
60.13
0.016
-0.012
CD
12.61
-59.31
0.011
-0.008
DE
-94.90
-88.50
0.023
-0.016
EA
-27.60
16.58
0.006
-0.004
(-0.05)
(0.070)
(-0.050)
Misc
(0.07)
Eadj
Nadj
Adjusted vector components
Side
E
N
dE
dN
Eadj
Nadj
AB
30.16
71.05
0.014
-0.010
30.146
71.060
BC
79.80
60.13
0.016
-0.012
79.784
60.142
CD
12.61
-59.31
0.011
-0.008
12.599
-59.302
DE
-94.90
-88.50
0.023
-0.016
-94.923
-88.484
EA
-27.60
16.58
0.006
-0.004
-27.606
16.584
(-0.05)
0.070
-0.050
Misc
(0.07)
(0.000)
(0.000)