TRAVERSE COMPUTATION

Lecture 3

SGU 1053
SURVEY COMPUTATION

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Traverse:

• A series of lines whose lengths and angular relationships have been measured.

Types:
Loop
• Closed: Starts and ends at same point (loop)
or starts at known point and ends at another
known point (link).

Link

• Open: Starts at known or unknown point and

ends at unknown point.
Link
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Steps to Computation:

1. Adjusting angles or directions to fixed geometric conditions

2. Determining preliminary azimuth (or bearings) of the traverse lines
3. Calculating departures and latitudes and adjust them for misclosures
4. Computing rectangular coordinates of the traverse stations
5. Calculating the lengths and azimuths of the traverse lines after adjustment

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 1. Balancing Angles:

Apply an average correction to each angle where observing conditions were

approximately the same at all stations. The correction is found by dividing the
total angular misclosure by the number of angles.

a=1000 44.3’
b=1010 35.1’
b c= 890 05.3’
c d= 170 11.9’
a e=2310 24.6’
d 5400 01.2'
e
(interior angles) (n 2) *180

T 5400 00.0'

Misclosures = 1.2’
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• Correction: 01.2’/5 = 0.24’
• Subtract correction from each angles

Angles Angle Multiple Corr. Round Succ. Diff. Adj. Angles

a 1000 44.3’ 0.24 0.2 0.2 1000 44.1’

b 1010 35.1’ 0.48 0.5 0.3 1010 34.8’
c 890 05.3’ 0.72 0.7 0.2 890 05.1’
d 170 11.9’ 0.96 1.0 0.3 170 11.6’
e 2310 24.6’ 1.20 1.2 0.2 2310 24.4’
5400 01.2’ 1.2’ 5400 00.0’

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 2. Determining Preliminary Azimuth (or Bearings):
But first: The difference between Bearings and Azimuth

Bearing:
• The bearing of a line is the direction of the line with respect to a given meridian.
• A bearing is indicated by the quadrant in which the line falls and the acute angle
that the line makes with the meridian in that quadrant.
• Observed bearings are those for which the actual bearing angles are measured,
while calculated bearings are those for which the bearing angles are indirectly
obtained by calculations.
• A true bearing is made with respect to the astronomic north reference meridian.
• A magnetic bearing is one whose reference meridian is the direction to the
magnetic poles.
• The location of the magnetic poles is constantly changing; therefore the magnetic
bearing between two points is not constant over time.
• The angle between a true meridian and a magnetic meridian at the same point is
called its magnetic declination.

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Azimuth:

• The azimuth of a line is its direction as given by the angle between the meridian
and the line, measured in a clockwise direction.
• Azimuths can be referenced from either the south point or the north point of a
meridian.
• Geodetic azimuths traditionally have been referenced to the south meridian
whereas grid azimuths are referenced to the north meridian.
• Assumed azimuths are often used for making maps and performing traverses, and
are determined in a clockwise direction from an assumed meridian.
• Assumed azimuths are sometimes referred to as "localized grid azimuths."
• Azimuths can be either observed or calculated.
• Calculated azimuths consist of adding to or subtracting field observed angles from
a known bearing or azimuth to determine a new bearing or azimuth.
• Azimuths will be determined as a line with a clockwise angle from the north or
south end of a true or assumed meridian.

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Guidelines:

• Bearings are recorded with respect to its primary direction, north or south, and
next the angle east or west.
• A bearing will never be listed with a value over 90 degrees (i.e. the bearing value
always will be between over 0 degrees and 90 degrees.
• Bearing angles are computed from a given azimuth depending on the quadrant in
which the azimuth lies.
• When the azimuth is in the first quadrant (0° to 90°), the bearing is equal to the
azimuth.
• When the azimuth is in the second quadrant (90° to 180°), the bearing is equal to
180° minus the azimuth.
• When the azimuth is in the third quadrant (180° to 270°), the bearing is equal to
the azimuth minus 180°.
• When the azimuth is in the fourth quadrant (270° to 360°), the bearing is equal to
360° minus the azimuth.
• Since the numerical values of the bearings repeat in each quadrant, the bearings
must be labeled to indicate which quadrant they are in. The label must indicate
whether the bearing angle is measured from the north or south line and whether it is
east or west of that line.
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 N
1st Quadrant
4th Quadrant
P

420 16’ 38” Bearing P = N 420 16’ 38” E

Azimuth P = 420 16’ 38”

W E

Bearing Q = S 320 24’ 46” E

320 24’ 46”
Q Azimuth Q = 1470 35’ 14”

3rd Quadrant 2nd Quadrant

S
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• You need a direction of at least one line

b
c
a
d
e

T
Azimuth AT = 234017.6’

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Calculating unknown azimuths: 3 possible cases
This angle is
–( i–1 + i – 180 ), or
180° i-1 i
N i
N
N
N

i-1 i
i i-1
i
i
i i
i
i i

(a) (b)
N i
i
N

i
i
i
RLB RL A BS FS
i-1
all all
(c)

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Calculating subsequent azimuth:

o
• Case (a): i= i-1+ i – 180 (i = 0, 1, 2, ...)

i-1 i
i
i
i

i
(a)
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•Case (b): when ( i-1+ i – 180o) < 0:
i – 180 ) + 360
o o
i = ( i-1 +

180° i-1 i

N i
N

i-1
i

i
i
i

(b)
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Case (c): when ( i-1+ I – 180o) > 360o:
i = ( i-1+ i – 180 ) – 360
o o

(b)
N i
i
N

i
i
i

i-1

(c)

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 Line Computation Prelim. Prelim.
Azimuth Bearing
AB 2340 17.6’ + 1510 52.4’ - 3600 260 10.0’ N260 10.0’ E
BC 260 10.0’ + (1800 - 1010 34.8’) 1040 35.2’ S750 24.8’ E
CD 1040 35.2’ + (1800 - 890 05.1’) 1950 30.1’ S150 30.1’ W
DE 1950 30.1’ + (1800 - 170 11.6’) 3580 18.5’ N010 41.5’ W
EA 3580 18.5’ + (1800 - 2310 44.4’) 3060 54.1’ N260 10.0’ W

Azimuth AB
a=1000 44.3’

2340 17.6’

T
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3. Calculating Departures (X/E) and Latitudes (Y/N):

N
Departure
B

Latitude Direction
Length

A
LatAB = LAB x Cos(DirAB )
East Dep. and North Lat. is +ve
DepAB = LAB x Sin(DirAB ) West Dep. and South Lat. is -ve

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• If all angles and distances are measured perfectly, the of latitude and of
departures will be zero
• But errors exist.
• The difference is called departure misclosure and latitude misclosures
• The misclosure will give the indication of precision of measured angles and
distances

Linear Misclosure:
• Start at a but end up at a’:
b
c
a
d
a’ e

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Linear misclosure = 2 2
departuremisclosure latitude misclosure

linear misclosure
Relative precision =
traverselength

Order Max Max Typical survey task

Linear Misclosure Relative Precision
First 1 in 25000 Control or monitoring surveys
2 n

Second 1 in 10000 Engineering surveys;

10 n setting out

Third 1 in 5000
30 n
Fourth 1 in 2000 Surveys over small sites
60 n

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 Station Prelim. Bearing Length Departure Latitude
A N260 10.0’ E 285.10 +125.72 +255.88
B S750 24.8’ E 610.45 +590.77 -153.74
C S150 30.1’ W 720.48 -192.56 -694.27
D N010 41.5’ W 203.00 -5.99 +202.91
E N260 10.0’ W 647.02 -517.40 +388.50
A 2466.05 +0.54 -0.72

DepAB = LAB x Sin(DirAB ) = 285.10 x sin (260 10.0’ ) = +125.72

2 2
Linear misclosures = departuremisclosure latitude misclosure

2 2
= 0.54 0.72 = 0.90 meter
linear misclosure 0.90 1
Relative precision = = =
traverselength 2466.05 2700
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Traverse Computations and Adjustments
• The linear misclosures must be adjusted (distributed)
• There are a number of methods available for adjusting traverses.
• The most common are listed below:

a. Crandall Rule.
•The Crandall rule is used when the angular measurements (directions) are
believed to have greater precision than the linear measurements (distances).
•This method allows for the weighting of measurements and has properties
similar to the method of least squares adjustment.
•Although the technique provides adequate results, it is seldom utilized
because of its complexity.
•In addition, modern distance measuring equipment and electronic total
stations provide distance and angular measurements with roughly equal
precision.
•Also, a standard Least Squares adjustment can be performed with the same
amount of effort.

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b. Compass Rule (Bowditch Method)

• The Compass Rule adjustment is used when the angular and linear
measurements are of equal precision.
• This is the most widely used traverse adjustment method.
• Since the angular and linear precision are considered equivalent, the angular
error is distributed equally throughout the traverse.
• For example, the sum of the interior angles of a five-sided traverse should equal
540º 00' 00".0, but if the sum of the measured angles equals 540° 01' 00".0, a
value of 12".0 must be subtracted from each observed angle to balance the angles
within traverse.
• After balancing the angular error, the linear error is computed by determining
the sums of the north-south latitudes and east-west departures.
• The misclosure in latitude and departure is applied proportional to the distance
of each line in the traverse.

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c. Least Squares

• The method of least squares is the procedure of adjusting a set of observations

that constitute an over-determined model (redundancy > 0).
• A least squares adjustment relates the mathematical (functional model) and
stochastic (stochastic model) processes that influence or affect the observations.
• Stochastic refers to the statistical nature of observations or measurements.
• The least squares principle relies on the condition that the sum of the squares
of the residuals approaches a minimum.

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Traverse Adjustment via Compass (Bowditch Rule)

• The Compass Rule is a simple method and is most commonly employed for
engineering, construction, and boundary surveys.
• It is also recognized as the accepted adjustment method in some state minimum
technical standards.

• Calculate latitudes (dY or dN) and departures (dX or dE) correction of the
traverse misclosure:
(total departuremisclosures)
dE x length AB
traverse parameter

(total latitude misclosures)

dN x length AB
traverse parameter

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• Distribute the misclosure latitudes and departures over the traverse
• Compute adjusted coordinates of the traverse stations
• Calculate final adjusted lengths and azimuths between traverse points

Line Prelim. Length Departure Latitude Balanced

Azimuth East West North South Dep. Lat.
AB 260 10.0’ 285.10 (-0.06) (+0.08)
+125.72 255.88 +125.66 +255.96
BC 1040 35.2’ 610.45 (-0.13) (+0.18)
+590.77 153.74 +590.64 -153.56
CD 1950 30.1’ 720.48 (-0.16) (+0.21)
192.56 694.27 -192.72 -694.06
DE 3580 18.5’ 203.00 (-0.05) (+0.06)
5.99 202.91 -6.04 +202.97
EA 3060 54.1’ 647.02 (-0.14) (+0.19)
517.40 388.50 -517.54 +388.69
+716.49 -715.95 +847.29 -848.01
misclosure +0.54 misclosure -0.72

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4. Computing Rectangular Coordinates:

• You need coordinates to compute:

• Length and directions of lines
• Area XB = XA + departure AB
• Curve calculations YB = YA + latitude AB
• Locating inaccessible points
• Ease in plotting maps

Point Coordinates
X (Easting) Y (Northing)
A 10000.00 10000.00

B 10125.66 10255.96

C 10716.30 10102.40

D 10523.58 9408.34

E 10517.54 9611.31

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Computing Lengths and Azimuth/Bearings from Departure/Latitudes:
departure
tan azimuth (or bearing)
latitude

departure latitude
length
sin azimuth (or bearing) cos azimuth (or bearing)
2 2
departure latitude

• The process of computing lengths and directions from departures

and latitudes or from coordinates is called inversing

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5. Calculating the lengths and azimuths of the traverse lines
after adjustment:

• Used the adjusted departures and latitudes to compute adjusted azimuth and
lengths

Line Balanced
Length Azimuth
AB 285.14 26008.9'
BC 610.28 104034.4'
CD 720.32 195031.1'
DE 203.06 358017.6'
EA 647.25 306054.5'
125.66
tan azimuth AB 0.490936; azimuth AB 260 08.9'
255.96
2 2
length AB 125.66 255.96 285.14 meter

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