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TRAVERSING
Computations
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Traversing - Computations
Traverse computations are concerned with deriving co-ordinates for
the new points that were measured, along with some quantifiable
measure for the accuracy of these positions.
The co-ordinate system most commonly used is a grid based
rectangular orthogonal system of eastings (X) and northings (Y).
Traverse computations are cumulative in nature, starting from a
fixed point or known line, and all of the other directions or positions
determined from this reference.
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Traversing - Computations
Angle/Bearing Computations and Balancing
If angles are measured within a traverse, they need to be converted
to bearings (relative to the meridian being used) in order to be used
in the traverse computation.Before the bearings and azimuths are computed, the measured
angles are checked for consistency and to detect any blunders.
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Traversing - ComputationsFor closed traverses, a check can be applied to ensure that the
measured angles can meet the required specifications. For a
closed loop traverse with n internal angles, the check that is
used is:
7(internal angles) = (n 2) 180r
or
7(external angles) = (n + 2) 180r
For a closed link traverse, the check is given by
A1 +7(angles) A2 = (n 1) 180r
whereA1 is the initial or starting azimuth,A2 is the closing
or final azimuth, and n is the number of angles measured.
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Traversing - Computations
The numerical difference between the computed checks and themeasured sums is called the angular misclosure. There is usually a
permissible or allowable limit for this misclosure, depending upon
the accuracy requirements and specifications of the survey. A
typical computation for the allowable misclosure Zis given by
Z= kn
where n is the number of angles measured and k is a fractionbased on the least division of the theodolite scale. For example,
if k is 1', for a traverse with 9 measured angles, the allowable
misclosure is 3 '.
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Traversing - ComputationsOnce the traverse angles are within allowable range, the remaining
misclosure is distributed amongst the angles. This process is called
balancing the angles :
(i) arbitrary adjustment if misclosure is small, then it may be
inserted into any angle arbitrarily (usually one that may be
suspect). If no angle suspect, then it can be inserted into more
than one angle.
(ii) average adjustment misclosure is divided by number of
angles and correction inserted into all of the angles. (most
common technique)
(iii) adjustment based on measuring conditions if a line has
particular obstruction that may have affected observations,
misclosure may be divided and inserted into the two angles
affected.
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Traversing - Computations
Errors in angular measurement are not related to the size of the
angle.
Once the angles have been balanced, they can be used to compute
the azimuths of the lines in the traverse.
Starting from the azimuth of the original fixed control line, the
internal or clockwise measured angles are used to compute the
forward azimuths of the new lines.
The azimuth of this line is then used to compute the azimuth of thenext line and so on.
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Traversing - Computations
The general formula that is used to compute the azimuths is:
forward azimuth of line = back azimuth of previous line +clockwise (internal) angle
The back azimuth of a line is computed from
back azimuth = forward azimuth s180r
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Traversing - Computations
Therefore for a traverse from points 1 to 2 to 3 to 4 to 5, if the anglesmeasured at 2, 3 and 4 are 100r, 210r, and 190rrespectively, and theazimuth of the line from 1 to 2 is given as 160r, then
Az23 = Az21 + angle at 2 = (160r+180r) + 100r= 440r | 80r
Az34 = Az32 + angle at 3 = (80r+180r) +210r= 470r |110r
Az45 = Az43 + angle at 4 = (110r+180r) +190r= 480r |120r
1
2
3
4
5
100r210r
190r
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Traversing - Computations
Once all of the azimuths have been computed, they can be
checked and used for the co-ordinate computations.
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Co-ordinate Computations
From Angle Azimuth Dist E N Easting Northing To
1043.87 5492.91 A
A 172 39 ' 34.15 4.37 -33.87 1048.24 5459.04 E
E 118 34 ' 111 13 ' 50.30 46.89 -18.20 1095.13 5440.84 D
D 113 05 ' 44 18 ' 55.19 38.54 39.50 1133.67 5480.34 C
C 104 42 ' 329 00 ' 41.81 -21.53 35.84 1112.14 5516.18 B
B 102 11 ' 251 11 ' 72.11 -68.26 -23.26 1043.88 5492.92 A
A 101 28 ' 172 39 ' E
=253.56 =0.01 =0.01 Diff=
+0.01
Diff=
+0.01
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From Angle Azimuth Dist E N Easting Northing To
A 1043.87 5492.91 A
172 39 ' 34.15 4.37 -33.87
E 118 34 ' 1048.24 5459.04 E
111 13 ' 50.30 46.89 -18.20
D 113 05 ' 1095.13 5440.84 D
44 18 ' 55.19 38.54 39.50
C 104 42 ' 1133.67 5480.34 C
329 00 ' 41.81 -21.53 35.84
B 102 11 ' 1112.14 5516.18 B
251 11 ' 72.11 -68.26 -23.26
A 101 28 ' 1043.88 5492.92 A
172 39 ' =253.56 =0.01 =0.01 1043.87 5492.91
E Diff= +0.01 Diff=+0.01
E
Alternative layout
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Traversing - Computations
A
E
B
C
D
118 34 ' 113 05 '
104 42 '
102 11 '
172 39 '
352 39 '
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Misclosures and Adjustments
For closed traverses, since the co-ordinates of the final endingstation are known, this provides a mathematical check on the
computation of the co-ordinates for all of the other points. If the
final computed eastings and northings are compared to the known
eastings and northings for the closing station, then co-ordinate
misclosures can be determined. The easting misclosure xE isgiven by
xE= final computed easting final known easting
similarly, the northing misclosure xN is given by
xN= final computed northing final known northing
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Linear Misclosure
These discrepancies represent the difference on the ground
between the position of the point computed from the observations
and the known position of the point.
The easting and northing misclosures are combined to give the
linear misclosure of the traverse, where
linear misclosure = (xE2 + xN2)
xE
xN
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Traversing Precision
By itself the linear misclosure only gives a measure of how far thecomputed position is from the actual position (accuracy of the
traverse measurements).
Another parameter that is used to provide an indication of the
relative accuracy of the traverse is theproportional linearmisclosure.
Here, the linear misclosure is divided by total distance measured,
and this figure is expressed as a ratio e.g. 1 : 10000.
In the example given, if the total distance measured along a traverse
is 253.56m, and the linear misclosure is 0.01m, then the proportional
linear misclosure is
0.01/253.56 = 1/25356 or approximately 1 : 25000
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Traversing Angular Error
The required accuracy of the survey in terms of its proportional
linear misclosure also defines the equipment and allowable
misclosure values.
For example, for a traverse with an accuracy of better than 1/5000
would require a distance measurement technique better than1/5000, and an angular error that is consistent with this figure.
If the accuracy is restricted to 1/5000, then the maximum angular
error is
1/5000 = tanU
U = 0r00'41"
xE
xNU
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Traversing Angular ErrorThe angular measurement for each angle should therefore be better
than 0r00'41". The general relationship between the linear andangular error is given by the following table
Prop. Linear accuracy Maximum angular error Least count of instrument
1/1000 0 03' 26" 01'
1/3000 0 01' 09" 01'
1/5000 0 00' 41" 30"
1/7500 0 00' 28" 20"
1/10000 0 00' 21" 20"
1/20000 0 00' 10" 10"
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Traversing Angular Error
The maximum allowable error in the traverse which is given by
Z= kn
k therefore depends on the maximum allowable angular error as it
relates to the least count of the instrument.
For a 1/5000 traverse, the value ofk = 30", so Z= 30"n.
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Traversing - Computations
If a misclosure exists, then the figure computed is not
mathematically closed.
This can be clearly illustrated with a closed loop traverse.
The co-ordinates of a traverse are therefore adjusted for thepurpose of providing a mathematically closed figure while at the
same time yielding the best estimates for the horizontal positions
for all of the traverse stations.
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Traversing - Adjustments
There are several methods that are used to adjust or balancetraverses;
1. (i) Arbitrary method
2. (iii)L
east-Squares3. (iv) Transit rule
4. (v) Bowditch or Compass rule
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Adjustments - Arbitrary
The arbitrary method is based upon the surveyors individualjudgement considering the measurement conditions.
The Least Squares method is a rigorous technique that is founded
upon probabilistic theory.
It requires an over-determined solution (redundant measurements)
to compute the best estimated position for each of the traverse
stations.
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Adjustments Transit Rule
The transit rule applies adjustments proportional to the size of theeasting or northing component between two stations and the sum of
the easting and northing differences.
Therefore for two stations A and B, the correction to the easting and
northings differences (Eab and (Nab are given by;
correction to (Eab = xEy((Eab/(E)
correction to (Nab = xNy((Nab/(N)
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Adjustments Transit Rule
In this method, if a line has no easting difference, then it will not
have an easting correction, and similarly, if it has no northing
difference there is no northing correction.
Conversely, lines with larger easting and northing differences will
have larger corrections.
For example, consider a traverse that has an easting misclosure xEof 0.170m and a northing misclosure xN of 0.361m and the eastingand northing differences are 54.439m and 1.230m respectively. If
the sum of the easting differences is 587.463m and the sum of the
northing differences is 672.835m, then
correction to (Eab = 0.170y(54.493/587.643) = 0.016m
correction to (Nab = 0.361y(1.230/672.835) = 0.001m
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Adjustments Compass Rule
The Bowditch or Compass rule also applies a proportional adjustment,but in this case, the distances between the stations are used in proportion
to the total distance of the traverse. The corrections are given by
correction to (Eab = xEy(distanceab/total distance oftraverse)
correction to (Nab = xNy(distanceab/total distance of
traverse
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Adjustments Compass Rule
This is the most commonly used technique for adjusting traverses.Using the above example, if the distance between A and B was
67.918m, and the total distance of the traverse was 1762.301m, then
the corrections to be applied are
correction to (Eab = 0.170y(67.918/1762.301) = 0.006m
correction to (Nab = 0.361y(67.918/1762.301) = 0.014m
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Traversing - Computations
2.1.Blunder Detection
Since traverse measurements involve angular and distance
measurements, it is possible for blunders to exist in the
measurements that are not detected until the final co-ordinate
computations are made.
Angular blunders manifest themselves in the angular closure and
distance blunders in the co-ordinate closure, provided that the
traverse is properly closed.
In both cases it is possible to localise the blunder.
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Traversing - Computations
To find an angular blunder, the traverse is computed withoutdistributing the angular errors first in the forward direction, and then
in the reverse direction.
The point of intersection (where the co-ordinates are virtually the
same) between the forward and reverse computations represents thelocation where the angular blunder was made, provided that only one
blunder was made.
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Traversing - Computations
A distance blunder causes a shift in the traverse section in thedirection of the incorrect length.
This is detected by checking the size and direction of the linear
misclosure.
If the linear misclosure is near a round figure (e.g. 1m or 5m) then
a blunder probably exists within the measurements. The azimuth
of the misclosure is then computed by
Az = tan-1(xE/xN)
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Traversing - Computations
If the azimuth is similar to any of the traverse legs, then it is
likely that the distance blunder occurred when measuring this
leg, and it can be corrected by remeasuring the line.
If in the above example, the distance from A to B was measured
as 75.11, then the resulting values for xE and xN would be 2.83m and 0.96m respectively.
The azimuth of the misclosure would then be
Az = tan-1(-2.83/-0.96) = tan-1 (2.9479) = 251r15
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Traversing - Computations
This azimuth is almost the same as the azimuth of the line BA,
so the distance blunder has been detected in this line. Thismethod is limited when there are several legs with nearly the
same azimuth.