Lecture Traverse Comps

8/8/2019 Lecture Traverse Comps

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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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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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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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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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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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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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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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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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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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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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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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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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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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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.

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Lecture Traverse Comps