‫ اﻟﻤﻀﻠﻊ‬The
Civil Engineering Students
Year (1)
Second semester – Phase II
Traverse
Dr. Kamal M. Ahmed
Surveying Measurements
• Surveyors, regardless of how complicated the
technology, measure two quantities: angle and
distances.
• They do two things: map “ ‫ “رﻓﻊ‬or set-out “‫“ﺗﻮﻗﻴﻊ‬
• Angles are measured in horizontal or vertical planes
only to produce horizontal angles and vertical angles.
• Distances are measured in the horizontal, the vertical,
or sloped ‫ﻡﺎﺋﻠﺔ‬directions.
• Our calculations are usually in a horizontal or a
vertical plane for simplicity, sloped ‫ ﻡﺎﺋﻠﺔ‬values can be
calculated if needed.
• For example: maps are horizontal projections of
data, distances are horizontal on a map and so
are the angles.
• Assume that you are given the horizontal
coordinates X (E), and Y (N) of two points A
and B: (20,20) and (30, 40). If you measure the
horizontal angle CBA and the horizontal
distance AC, found them to be: 110° and 15m,
then the coordinates of C can easily be
computed, here is how:
C
• Calculate the azimuth of AB, then BC
• Calculate (∆E, ∆N) for BC
• Calculate (E, N) for C
B
A
• But, if you were given a slope distance or a
slope angle, you won’t be able to compute the
location (Coordinates) of C.
• What we did was to map “‫ ”ﻧﺮﻓﻊ‬point C, we
found out its coordinates, now you plot it on a
piece of paper, a “map” is a large number of
points such as C, a building is four points, and
so on.
• Now, if point C was a column “‫ ”ﻋﻤﻮد‬and we
wanted to set it out “‫”ﻧﻮﻗﻌﻬﺎ‬, then we know the
coordinates of C from the map:
• Calculate the angle ABC and the length of BC
• Setup the instrument “‫ ”اﻟﺠﻬﺎز‬,such as a theodolite, on B,
aim “‫ ”وﺝﺔ‬at A
• Rotate the instrument the angle ABC, measure a distance
BC, mark the point.
• You set out a point, then you can set out a project.
• In both cases, you need two known points such as A
and B to map or set out point C
• We call precisely known points such as A and B
“control points”
• In horizontal, we do a traverse “‫ ”ﻡﻀﻠﻊ‬to construct
new control points based on given points.
• You need at least two points given in horizontal ( or
one and direction) and one in vertical to begin your
project
• Definition:
Traversing
– A traverse is a series of consecutive lines whose
lengths and directions have been measured.
• Why?
– The purpose of establishing a traverse is to extend
the horizontal control. A survey usually begins with
one given vertical control and two ( or one and
direction) given in horizontal
– You need more than two points to control the
project, have enough known points to map any
point, and set-out any object any where in a large
project.
Procedure ‫ﺧﻄﻮات ﻋﻤﻞ اﻟﻤﻀﻠﻊ‬
Assume that you wanted to
map “calculate coordinates
of the building, trees, and
the fence “‫ ”ﺳﻮر‬in the
drawing, you are given
points A and B only, cannot
measure angle and
distance to corner F or the
trees!!
A
E
F
B
Grass
C
• You need to construct new control points “points of
known precise coordinates” such as C, D, and E to
measure from.
• You do that with a traverse
D
Procedure ‫ﺧﻄﻮات ﻋﻤﻞ اﻟﻤﻀﻠﻊ‬
1. Walk around and decide
which are the best locationsA
to have new control points
2. Construct the points, nails
on asphalt, concrete and
bolts, etc.
3. Measure all the angles and B
all the lengths of the
traverse
4. Check if the angles and
lengths are accepted
5. If rejected, re-do the work
6. If accepted, adjust the errors
and compute coordinates.
E
D
Grass
C
Coordinate Computations
• Assume that we were given a site to map, and
the coordinates of one point (A), and the
azimuth of the line (AB), we need more known
(control stations)
• We marked three more points around the site,
the four points make rectangle (or a square).
• We then measured all the internal angles and
the length of all the sides (lines).
• Using the given azimuth of AB and all angles,
we computed the azimuth of all the sides, we
get the following table:
Coordinate Computations
Point
Line
Azimuth
∆E =
∆N =
Length
(α) d sin( α)
d cos( α)
A
N
200.00 350.00
AB
100.10
0° 00' 00''
0.00
100.10
B
200.00 450.10
BC
100.00 90° 00' 00" 100.00
0.00
C
300.00 450.10
CD
100.00 180°00'00" 0.00
-100.00
D
300.00 350.10
DA
99.70 270°00'00”
- 99.70
0.00
A
Sum
E
200.30 350.10
399.80
0.30
0.10
Questions
Note that the coordinates of A when computed at the bottom of the table,
are not the same as given coordinates.
Also note the relationship between that error and the sum of Northings
and Eastings.
How do you explain that?
Assume that the traverse was a perfect square of 100 m side length and
oriented towards the north, what you notice in our measurements, and
how can you relate that to the error in A?
The concept of Linear Closing Error
B
c
Closing error = 0.32m
Notice that if the
corrections are ignored,
the value of the errors
will
Appear when you recompute the coordinates
of the first point (A).
Σ∆N=0.1
D
A
Σ∆E=0.3
The concept of Angular Closing Error
• Now let us assume the same perfect square of 90°
angles and 100 m sides. When measured all lengths
were correct, while the angle at B was in error by 10°
when measured, the surveyor reported 100°. What
happens?
B
c
Here is the
perfect traverse
that we are
trying to
measure:
A
D
The concept of Closing Error
• Error in angles OR error in distances will
result in a closing error: last point will not be
at the first point.
• The problem is that we do not know where
the errors are and how much each error?
• Measurements are never exact, we always
assume that we have errors in angles and
distances.
• Before we learn how to compute the errors
and how to adjust for them, let us learn some
issues with traverse
Closed and Open Traverses
• A closed traverse is the one that starts and
ends at known points and directions,
whether the shape is closed or not
• A closed traverse can be a polygon
{closed shape} or Link {closed geometryopen shape
Closed (polygon or link)
traverses
Link
Polygon
Open TRAVERSE
∑∆XObserved
L3
L4
L5
B
MX
•Open Traverses are not used in engineering
control applications, why?
•The problem: there is no way to check the for
the errors; you will have to accept whatever
coordinates computed.
∆Y=YB-YA
L2
XY
L1
∆X=XB-XA
M
∑∆YObserved
A
Observed Location
MY
True Location
Traverse Notations
• We will only cover the closed Traverse with interior
angles measured.
Traverse Stations
• Successive stations should be inter visible.
• Stations are chosen in safe, easy to access
places.
• Lines should be as long as possible
– To reduce the number of lines
– Short lines will produce less accurate angles,
the traverse gets distorted as shown below.
T1
A
T3
T2
B
T4
Traverse Stations
• Angles should be as equal as possible and better
be 30 to 150°, why????
• Lines should be and as equal as possible, Why?
• Stations must be referenced to retrieve them if lost.
We produce a “descriptive card” for each point
Descriptive card for a
traverse point
‫اﻟﺸﻤﺎل‬
‫‪ .3‬ﻋﻤل ﻜﺭﻭﻜﻴﺎﺕ ﺍﻟﻨﻘﻁ‬
‫أ‬
‫ﻃﺮﻳﻖ رﺋﻴﺴﻲ‬
‫ﻃﺮﻳﻖ أ‬
‫ب‬
‫هـ‬
‫أ‬
‫ﺣﺪﻳﻘﺔ‬
‫ﻣﺒﻨﻲ ﺳﻜﻨﻲ‬
‫وأداري‬
‫ﻃﺮﻳﻖ ب‬
‫روﺑﻴﺮ‬
‫)‪110.25‬م(‬
‫د‬
‫ج‬
‫ﺸﻜل )‪ (8.1‬ﻜﺭﻭﻜﻲ ﻋﺎﻡ ﻟﻠﻤﻨﻁﻘﺔ‬
‫ﺱﻠﻢ ﻡﺒﻨﻲ‬
‫ﺵﺠﺮة‬
‫‪ 12.73‬م‬
‫‪ 8.67‬م‬
‫ﺸﻜل )‪ (8.2‬ﻜﺭﻭﻜﻲ )ﻜﺭﺕ ﻭﺼﻑ( ﻨﻘﻁﺔ ﺝ‬
‫‪ 11.24‬م‬
‫ﻋﻤﻮد أﻥﺎرﻩ‬
‫ﺘﺜﺒﻴﺕ ﻨﻘﻁ ﺍﻟﻤﻀﻠﻊ‬
‫•‬
‫•‬
‫•‬
‫•‬
‫‪:‬‬
‫ﺘﺜﺒﺕ ﻨﻘﻁ ﺍﻟﻤﻀﻠﻊ ﺃﻤﺎ ﺒﻭﺘﺩ ﺨﺸﺒﻲ ﺃﻭ ﺯﺍﻭﻴﺔ ﺤﺩﻴﺩﻴﺔ ﻓﻲ ﺍﻷﺭﺍﻀﻲ ﺍﻟﺯﺭﺍﻋﻴﺔ‬
‫ﺃﻭ ﻤﺴﻤﺎﺭ ﺤﺩﻴﺩ ﻓﻲ ﺍﻟﻤﻨﺎﻁﻕ ﺍﻟﺼﺨﺭﻴﺔ ﻭﺍﻹﺴﻔﻠﺘﻴﺔ ﺃ‬
‫ﻭ ﺒﻘﺎﻋﺩﺓ ﺨﺭﺴﺎﻨﻴﺔ ﻭﺯﺍﻭﻴﺔ ﺤﺩﻴﺩﻴﺔ ﻓﻲ ﺍﻟﻤﻨﺎﻁﻕ ﺍﻟﺭﻤﻠﻴﺔ‬
‫ﻭﻴﺴﺘﺤﺴﻥ ﻭﻀﻊ ﻋﻼﻤﺔ ﺒﺎﻟﺒﻭﻴﺔ ﻜﻠﻤﺎ ﺃﻤﻜﻥ ﻋﻠﻰ ﺍﻟﻨﻘﻁ ﻴﻭﻀﺢ ﺭﻗﻤﻬﺎ ﻭﺘﺎﺭﻴﺦ‬
‫ﺘﺜﺒﻴﺘﻬﺎ ‪.‬‬
Traversing by Interior Angles
• All internal angles and all horizontal distances are
measured
• Each angle is measured in direct and reverse,
• Each angle is observed at least three times.
• A line of known direction should either be given or
assumed, what is a line with known direction?
• If the line of known direction is not a member of the
traverse, the angle to a traverse member should be
measured. Why?
Closing Error In Traverses
• We measure two values: angles and
distances
• Because of errors in both measurements,
we get angle misclosure and linear
misclosure ( closing error).
• Both types of errors result in error in
closure, we need a way to separate the
error of angles from the error in distances
to check and adjust them separately
The concept of Angle Misclosure
• Here is how the measured traverse will look:
Line AB was
correct
B
c
Line BC was
correct, but
angle A was
wrong
The rest of the
lines and angles
are correct
A
A’
D
Computations and Adjustments
of Angle Misclosure
• The sum of internal angles of a polygon of (n) points =
(n - 2) * 180o
• Angle misclosure = difference between the sum of the
measured angles and the geometrically correct total
for the polygon.
• The misclosure is divided equally among the readings
keeping in mind the measuring accuracy, and should
be done at the beginning of the adjustment.
Judging The Angle Misclosure
• Usually the standards give an equation of the form:
• Max allowed angle misclosure c = k * √n
where (n)
is the number of points and K is a constant defined
according to which standards used
• If angles are accepted, correct by dividing the error
equally among the angles
• For example: The Federal Geodetic Control
Subcommittee: 1.7, 3, 4.5, 10, and 12” for first-order,
second-order class I, second-order class II, third-order
class I, third-order class II
• If no standards were given, then assume that:
Allowed angle misclosure = 3σ√n =
3 * theodolite accuracy * √ No. of angles
Example (1)
For 5 points traverse, the sum of internal angles is
538° 40’ and the accuracy of the measured angles is
10’. Do you accept the observations?.
Answer
• Ang. Clos. Err. = 538 40 – 180 x (5 – 2)
= 538 40 – 540 = - 1 20 = - 80’
• Allowable Angle Closing Error = 3 x 10’ √5 = 67’
• Reject the observations. You have to re-observe
Example (2)
If the angles were observed to the nearest 30” in 5
points traverse. If the angular closing error was 2’,
correct the angles.
Answer
Allowable angular closing error = 3 x 30” √ 5 = 201”
As ∆θ = 120” (≤ 201” ) , accepted.
Correction in each angle = - 120 “ / 5 = - 24”
But the angles were observed to the nearest 30”, the
corrections will be :
- 30”, - 30”, - 30” , - 30” & 0 .
The concept of Linear Closing Error
B
c
Assume that the
traverse in reallity
was a perfect
square.
A’
Σ∆N
A
D
Σ∆E
- A will close at A’,
- AA’ is the linear closing
error
Assume that
there was an
error in
measuring the
length AB only, all
other lengths and
angles were
correct
N
EDA
ECD
- ve
- ve
D
C
A
B
If the traverse is
closed, then
Σ∆E = 0
Σ∆N = 0
and
EAB
EBC
+ ve
+ ve
E
N
EDA
ECD
- ve
- ve
D
Σ∆N
C
Σ∆E
If the traverse is
closed, then
Σ∆E = 0
A’
A
and
B
Σ∆N = 0
If the traverse is not closed,
Then Σ∆E = Ec
EAB
EBC
+ ve
+ ve
and Σ∆N = Nc
E
Computations of Linear Closing
Error
• If he closing error is (W) then
Ew = Σ∆E and
Nw = Σ∆N,
W = length of closing error = √ Ew2 + Nw2
Fractional Closing error = traverse precision =
W/ΣL
Direction of the error = Azimuth =
tan-1 (Ew / Nw) = tan-1 (Σ∆E / Σ∆N)
Adjustment of Linear Misclosure
• Compute and adjust the angle misclosure
• Compute the linear misclosure:
– Compute the azimuth of a traverse side
– Compute the azimuth of all the sides
– Compute the departure and latitude of all the sides
– Compute the Misclosure in (E) direction =
sum of the departures.
– Compute the Misclosure in (N) direction =
sum of the latitudes.
– Compute the linear misclosure
– If accepted, use the Compass (Bowditch) rule to
adjust:
Compass (Bowditch) Rule
(
Σ ∆E
ΣL
) (L )
(
Σ ∆N
ΣL
) (L )
Correction in departure for AB = Correction in latitude for AB = -
AB
AB
Where:
L is the length of a line, and (Σ L) is the perimeter
Computations of Coordinates
• Add the corrections to the departure or the
latitude of each line to get the adjusted
departure or latitude
• Compute the adjusted point coordinates
using the corrected departure or latitude:
Ei = E i-1 + ∆E
Ni = N i-1 + ∆N
• Check that the misclosure is zero.
Example of Standards
Horizontal Control Accuracy Standards For Traverse
(By The Federal Geodetic Control Subcommittee (FGCS))
1st
2 nd
3 rd
Order
Class
Angular
Closure
1.7”√n
I
II
I
II
3.0”√n
4.5”√n
10.0”√n
12.0”√n
Linear 0.04√∑L 0.08√∑L 0.20√∑L 0.40√∑L 0.80√∑L
or,
or,
or,
or,
or,
Closure
1/100,00 1/50,000 1/20,000 1/10,000 1/5,000
(after
angul.
0
adj.)
Acceptable Relative precision
1 in 5000 & 30” √n for most engineering surveys
1 in 10000 & 10” √n for control, for large projects
1 in 20000 & 2” √n for major works and monitoring for
structural deformation etc.
Ex.: The angles were observed to the nearest 30” in 5
points traverse. If the angular closing error was 2’,
correct the angles.
Answer
Allowable angular closing error = 3 x 30” √ 5 = 201”
As ∆θ = 120” (≤ 201” ) , accepted.
Correction in each angle = - 120 “ / 5 = - 24”
But the angles were observed to the nearest 30”, the
corrections will be :
- 30”, - 30”, - 30” , - 30” & 0 .
point
Length L
Azimuth
AZ
Departure
L sin (Az)
Latitude
L cos (Az)
Correction
Departure
(WN/ΣL)* L
A
285.10
26 10.0’
125.72
255.88
610.45
104
35.2’
590.77
- 153.74
720.48
195
30.1’
- 192.56
- 694.27
203.00
358
18.5’
- 5.99
202.91
747.02
306
54.1’
- 517.40
388.5
WE =+0.54
WN =
- 0.72
B
C
D
E
A
Sum
P=2466.05
Latitude
(WE/ ΣL)* L
Balanced
Departure
E
Latitude
N
E
N
Other Methods
There are several methods that are used to adjust or
balance traverses;
1.
Arbitrary method
2.
Transit rule
3.
Least-Squares method
Traverse Area
D
E
C
B
A
Traverse area = 1 Σ { Ei (Ni+1 - Ni-1)}
2
•Multiply the X coordinate of each point by the difference
in Y between the following and the preceding points, half the sum
is the area
•The formula will work for traverses lettered in a clockwise
direction, but it will give a correct area with a negative sign.
•The formula should work if you switch the N and the E.