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Chapter (4)Chapter (4)

Traverse ComputationsTraverse ComputationsIntroductionIntroduction TheThe surveysurvey procedureprocedure knownknown asas traversingtraversing

isis f undamentalf undamental toto muchmuch surveysurveymeasurementmeasurement..

TheThe procedureprocedure consistsconsists of of usingusing aa varietyvariety of of instrumentinstrument combinationscombinations toto createcreate polarpolarvectorvector inin space,space, thatthat isis 'lines''lines' withwith aamagnitudemagnitude (distance)(distance) andand directiondirection(bearing)(bearing)..

TheseThese vectorsvectors areare generallygenerally contiguouscontiguous andandcreatecreate aa polygonpolygon whichwhich conf ormsconf orms toto variousvariousmathematicalmathematical andand geometricalgeometrical rulesrules (which(whichcancan bebe usedused toto checkcheck thethe f ieldworkf ieldwork andandcomputations)computations)..

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The equipment used generally consists of something to determine direction like a

compass or theodolite, and something todetermine distance like a tape orElectromagnetic Distance Meter (EDM).There are orderly field methods and

standardized booking procedures to minimizethe likelihood of mistakes, and routinemethods of data reduction again to reduce thepossible occurrence of errors.

The most fundamental of these checks is toperform a closed traverse that is a traversethat starts and finishes on either the samepoint or known points, (similar in concept to a

level run).

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The Function of Traverses

Traverses are normally perf ormedaround a parcel of land so that featureson the surf ace or the boundarydimensions can be determined.

Of ten the traverse stations will berevisited so that perhaps three-dimensional topographic data can beobtained, so that construction data can

be established on the ground.

A traverse provides a simple networkof 'known' points that can be used to

derive other inf ormation.

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Types of traverse

There are two types of traverse used in survey.These are open traverse, and closed traverse.

D1

A

B

C

1B

A C

D

2

A

1

D

B

C

Open Traverse

Closed Traverse

Link Traverse

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Open Traverse

An open traverse begins at a point of knowncontrol and ends at a station whose relativeposition is known only by computations.

The open traverse is considered to be the least

desirable type of traverse, because it provides nocheck on the accuracy of the starting control orthe accuracy of the fieldwork.

For this reason, traverse is never deliberately

left open.

Open traverse is used only open projects likeroads, cannels, railway lines, shorelineprotection.

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Closed Traverse

This traverse starts and ends at stations of known control.There are two types of closed traverse ±closed on the startingpoint and closed on a second known point.

Closed loop traverseThis type of closed traverse begins at a point of knowncontrol, moves through the various required unknown points,and returns to the same point.

This type of closed traverse is considered to be the secondbest and is used when both time for survey and limited surveycontrol are considerations.

It provides checks on fieldwork and computations andprovides a basis for comparison to determine the accuracy of the work performed.

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The first step in checking a closed traverse

is the addition of all angles.Interior angles are added and comparedto (n-2)*180o. Exterior angles are added and compared

to (n+2)*180o

. Deflection angle traverses arealgebraically added and compared to 360o. The allowable misclosure depends on the

instrument, the number of traverse stations,and the intention for the control survey.c =K * n 0.5

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wherec = allowable misclosure.

K = f raction of the least count of theinstrument, dependent on the number of repetitions and accuracy desired (typically 30"f or third-order and 60" f or f ourth-order)n = number of angles.

Exceeding this value, given the parameters, may indicate some other errors are present, of angulartype, in addition to the random error.The angular error is distributed in a manner suitedto the party chief before adjustment of latitude anddepartures. Adjustment of latitudes and departures isthe accepted method.The relative point closure is obtained by dividing theerror of closure (EC ) by the line lengths.

Relative point closure = EC / S of the distances

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Closed traverse between two known control points

This type of closed traverse begins from a point of known control, moves through the various requiredunknown points, and then ends at a second point of known control.

The point on which the survey is closed must be apoint established to an equal or higher order of accuracy than that of the starting point. This is thepreferred type of traverse.

It provides checks on fieldwork, computations, andstarting control. It also provides a basis forcomparison to determine the accuracy of the workperformed.

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The procedure for adjusting this type of

traverse begins with angular error just as ina loop traverse.To determine the angular error a formula isused to generalized the conversion of angles

into azimuth.The formula takes out the reciprocalazimuth used in the back sight as (n-1)stations used the back-azimuth as a back

sight in recording the angles.A1 + a1 + a2 + a3+ ...+ an -(n-1)(180o )=A2

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If the misclosure is exceeded, the angular

error may have been exceeded or thebeginning and ending azimuths are in error ororiented in different meridian alignments.If beginning and ending azimuths were

taken from two traverses, and the anglerepetitions were found to be at least an orderof magnitude better than the tabulatedangular error, the ending azimuths may

contain a constant error which may beremoved to improve the allowable error.GPS or astronomic observations may beused to find the discrepancy if the benefit of

this.

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Traverse f ieldwork

The easiest way of visualizing the traversingprocess is to consider it to be the formation of apolygon on the ground using standard surveyprocedures.

If the traverse is being measured using atheodolite (which is the normal case) then anglesare observed to survey stations on both faces for agiven number of rounds, and booked and reducedaccordingly.

The stations being observed are pre marked andtargeted with range poles or traversing targets, orsimply by a plumb-bob string for the duration of the angle measurement.

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If bearings are being observed with a

magnetic compass then care must be takento reduce the effect of variation indeclination over the period of the survey,and especially to avoid the effects of local

attraction.This is done by avoiding nearby metallicobjects, and by observing both forward andreverse bearings for each traverse line.

Whatever method is used for themeasurement of distance then allappropriate corrections should be made,and the distances reduced to horizontal.

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Choice of points

Planning - establish requirements for accuracy,density and location of control points.Reconnaissance - nature of terrain, access,location of points.

Station markingStation marking - type of mark, reference.Protection.Description Card.

ObservationsA

ngular and Distance Measurements.Angular Measurement ± Targets, Reading andbooking procedure.Linear Distance - Standard, slope, temp.Booking procedure.

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Traverse Computations

1 Angular Closure of Closed loop traverse

Using a theodolite we can measure all theinternal angles.The sum of the internal angles of a polygon(traverse) is given by the rule: = (n -2)* 180O

Where n is the number of sides of the traverse,and each internal angle. Any variation f rom this sum is known as themisclosure and must be accounted f or, either

through compensation (if it is an acceptableamount) or elimination by repetition of theobservations.An angular closure is computed f or traversesperf ormed with either Theodolites or magnetic

compasses.

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A larger misclosure could be expected whenusing a magnetic compass, but in any case it

must be calculated and removed.The Angular Misclosure = Measured Angles - Internal AnglesMaximum Angular Misclosure

=2*Accuracy of Theodolite *¥ (No. of Angles)Calculation of Whole Circle Bearing

When the angles is adjusted, then a bearing isadopted for one of the lines (or a known bearing is

used) and bearings for all the lines are computed.

The bearing of a line is computed by adding 180°to the bearing of the line before, and thensubtracting the included angle ().

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Example:

Observations, using a 6" Theodolite, were taken

in the f ield f or an anti - clockwise polygontraverse, A, B, C, D. The bearing of line AB is tobe assumed t o be 0o and the co-ordinates of station A are (3000.00 m E ; 4000.00 m N).

A

B

NC

D

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horizontalhorizontaldistancedistance

( m )( m )

linelineobservedobservedclockwiseclockwise

horizontal anglehorizontal angle

traversetraverse

stationstation

638638..5757ABAB132132ºº 1515

3030""

AA

15761576..2020BCBC126126ºº 12125454""

BB

38243824..1010CDCD069069ºº 41411818""CC

31333133..7272DADA031031ºº 50503030""

DD

360360ºº 0000

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Solution

Calculation of Angular misclosure (Internal Angles)= 360º 00 12"

(Internal Angles should be (N-2)*180º= 360º 00 00"

The Angular Misclosure()=

Measured Angles - Internal Angles= 360º 00 12" - 360º 00 00" = 12"

Allowable = 2 * 6" * ¥4 = 24" OK

Theref ore distribute errorThe correction / angle = -12/4= 3"

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correctedcorrectedhorizontal anglehorizontal angle

correctioncorrectionobserved clockwiseobserved clockwisehorizontal anglehorizontal angle

traversetraverse

stationstation

132132ºº 15152727""

--33""132132ºº 1515 3030""AA

126126ºº 1212

5151""

--33""126126ºº 1212 5454""BB

069069ºº 41411515""

--33""069069ºº 4141 1818""CC

031031ºº 50502727""--33""031031ºº 5050 3030""DD

360360ºº 00000000""

--1212""360360ºº 0000 1212""

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Calculation of Whole ± circle Bearing

bearingbearingcirclecirclewholewholebearing bearing backbacklineline

(W.C.B.)(W.C.B.)bearingbearingf orwardf orwardstationstationangleanglecorrectedcorrectedlineline

000000000000000000000000ABAB

00000000180180BABA

51511212126126BB

5151121230630651511212306306BCBC

51511212126126CBCB

151541416969CC

0606545419519506065454195195CDCD

060654541515DCDC

272750503131DD333344444747333344444747DADA

33334444227227ADAD

27271515132132AA

CheckCheckABAB

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Linear closureThe method of checking the distance component of

the closed traverse is known as performing a linearclosure.In its simplest form this consists of converting thecorrected angles into bearing and then computingthe partial Easting and Northing for each line.

Easting = D . Sin Northing = D . Cos These values are then summed, and any deviationfrom the expected value is assessed.In a traverse that starts and finishes on the samepoint the total change in position should be zero, andin a traverse that starts and finishes on points thathave a known position the sum should equal theknown displacement.

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An angular closure must be performed first, asthese formulae contain two measured variables(direction and distance) the bearings must havetheir error eliminated so we can attribute theremaining error to the distances.

)(*

)(*

1

11

1

11

Nn N Cos D N

En E Sin D E

i

i

n

i

n

i

i

i

n

i

n

i

!!(

!!(

§§§§

!!

!!

U

U

If the linear misclosure is acceptable, then thiscan be adjusted out of the network, but if themisclosure is too large then the fieldwork shouldbe repeated (unless the source of the problem canbe isolated).

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linear misclosure In above examplecan be calculated as f ollow

D cosD cos

( m )( m )

D sinD sin

(( m )m )

bearingbearingcirclecirclewholewholeDistanceDistance

( m )( m )lineline

++638638..575700..00000000""00000000ºº638638..5757ABAB

++931931..227227--12711271..7017015151""1212306306ºº15761576..2020BCBC

--36773677..764764--10471047..7547540606""5454195195ºº38243824..1010CDCD

++21072107..313313++23192319..3613613333""44444747ºº31333133..7272DADA

--00..654654--00..09409491729172..5959

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e = ¥ ( E2 + N2 )

= ¥ (0.0942

+ 0.6542

) = 0.661me is the LINEAR MISCLOSUR E

Fractional Linear Misclosure (FLM) = 1 in ( D / e )1 in (9172.59 / 0.661) = 1 in 13900

Acceptable FLM values :-

1 in 5000 f or most engineering surveys

1 in 10000 f or control f or large projects

1 in 20000 f or major works andmonitoring f or structural def ormation etc.

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ExampleConsider the f ollowing traverse and traverse table:

horizontalhorizontal

Distance (m)Distance (m)

linelineObserved Observed clockwiseclockwise

horizontal anglehorizontal angle

traversetraverse

stationstation

127127..5454ABAB096096ºº 5454 1010""AA

086086..3232BCBC107107ºº 3232 3030""BB

078078..4545CDCD141141ºº 2727 1010""CC

149149..6868DEDE087087ºº 1515 4040""DD096096..0202EAEA106106ºº 4949 4040""EE

538538..1111LL539539ºº 5959 1010""

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96o 54¶ 10´

127.54 m.107o 32¶ 30´

96.02 m.

78.45 m.

86.32 m.

149.68 m.

141o 27¶ 10´

106o 49¶ 40´87o 15¶ 40´

A

B

C

DE

N

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Solution

Calculation of Angular misclosure: (Internal Angles) = 539º 59 10" (Internal Angles) should be (n -2)*180 = (5-2)*180=540º 00 00"

The Angular Misclosure()

= Measured Angles - Internal Angles= 539º 59 10" - 540º 00 00" =- 50"

Allowable = 2 * 20" * ¥ 5= 89.44" OK

Therefore distribute error.The correction/angle = 50"/5= 10³

The angles area adjusted for this misclosure amount,this case 10 seconds would be added to each angle todistribute the misclosure evenly throughout thetraverse.

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W.C.B.W.C.B.corrected anglecorrected anglelineline

000000ºº 00000000""

(A)(A) 096096ºº 5454 2020""ABAB

072072ºº 27272020""

(B)(B) 107107ºº 3232 4040""BCBC

111111ºº 00000000""

(C)(C) 141141ºº 2727 2020""CDCD

203203ºº 44441010""

(D)(D) 087087ºº 1515 5050""DEDE

276276ºº 5454

2020""

(E)(E) 106106ºº 4949 5050""EAEA

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Linear closure

Latitu.Latitu.( m )( m )

Depar.Depar.( m )( m )

bearingbearingdistancedistance( m )( m )

lineline

127127..545400..000000000000ºº 00000000""

127127..5454ABAB

2626..0210218282..305305072072ºº 27272020""

086086..3232BCBC

--2828..1141147373..239239111111ºº 0000

0000""

078078..4545CDCD

--137137..018018--6060..250250203203ºº 44441010""

149149..6868DEDE

1111..545545--9595..323323276276ºº 5454

2020""

096096..0202EAEA

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From the table, E = -0.029, and N = -0.026This is then converted to a vector, expressing themisclosure in terms of a bearing and distance.

Distance =( E 2 + N2 )1/2 = 0.039 meters,Bearing = tan-1 ( E/ N) = 227° 30µThen the work is repeated. Conventionally the misclosure isexpressed as a ratio of the total perimeter of the traverse andreferred to as the 'accuracy'.

In this case this is 1:13,795 which satisfies requirements underthe Survey Coordination Act.If the misclosure was found to be large then it is likely that amistake had occurred during the field process.The bearing of the misclosure vector can be used as an indication

of the line in which the mistake occurred, however this is a guideonly.Naturally if the misclosure was close to one physical length of themeasuring device (say 50m) then it is likely that a chain lengthwas omitted somewhere. If the source of the mistake cannot beisolated,

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If the coordinates of point A ( 2000,5000 )

Now we will go to correct the coordinates of thepoints of the traverse

¨N¨N

corr.corr.

¨̈EE

corr.corr.

corr. forcorr. for

nortingnorting

corr. forcorr. for

eastingeasting

¨N¨N

uncorr.uncorr.

¨̈EE

uncorr.uncorr.

LineLine

127127..54654600..00700700..00600600..007007127127..545400..000000ABAB

2626..0250258282..31031000..00400400..0050052626..0210218282..305305BCBC

--2828..1101107373..24324300..00400400..004004--2828..1141147373..239239CDCD

--137137..011011--6060..24224200..00700700..008008--137137..018018--6060..250250DEDE

1111..550550--9595..31831800..00500500..0050051111..545545--9595..323323EAEA

000000..02602600..029029--00..026026--00..029029

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northing (m)northing (m)Easting (m)Easting (m)pointpoint

50005000..00000020002000..000000AA

51275127..5465462000.0072000.007BB

51535153..57157120822082..317317CC

51255125..46146121552155..560560DD

49884988..45045020952095..318318EE

50005000..00000020002000..000000AA

Final corrected coordinates:

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Traverses - Missing Data: As a rule traverses are always closed, either onto them

selves or between known points so that an estimate of accuracy and precision can be obtained, as well as acheck on our f ieldwork.

There are rare occasions where traverses cannot beclosed, and more commonly there are situations whereopen traverses run off a rigorous network are used todetermine the dimensions of features that are notreadily accessible.

The use of traversing procedures and calculation todetermine these dimensions is based on themathematics of a closed traverse.

That is, the data that is missing f rom the traverse ispresumed to be that which would close the traverse. If we adopt this procedure, then an additional

condition applying to our measurements is known

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0)(*

0)(*

1

11

1

11

!!!(

!!!(

§§

§§

!!

!!

N n N C os N

E n E Sin E

i

i

n

i

n

i

i

i

n

i

n

i

U

U

The missing elements of a traverse polygonthat can be solved for are as follows:

1.Bearing and Distance of One Line .2.Bearing of One Line, Distance of Another.3.Distance of two Lines.4.Bearing of two Lines.

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Example

I n a theodolite survey the f ollowing details

were noted and some of the observations weref ound to be missing.

Length ( m )Length ( m )W.C.B.W.C.B.lineline

480480060060ºº 0000 0000""ABAB11801180115115ºº 0000 0000""BCBC

??235235ºº 4040 0000""CDCD

12051205??DADA

Calculate the missing data ?

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Solution

latitudelatitudeDepartureDepartureR.B.R.B.Length( m )Length( m )lineline

++240240415415..6969NN 6060º Eº E480480ABAB

--498498..696910691069..4444SS 6565º Eº E11801180BCBC--00..564564 LL--00..826826 LLSS 5555ºº

4040WWLLCDCD

12051205 coscos 12051205 sinsin

12051205DA

DA

0000

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In Departure415.69 + 1069.44 ±0.826 L +1205 Sin () = 00.001205 Sin () = + 0.826 L ± 1485.13

1452025Sin2

()=0.682L2

±2453.43L+2205611.12 (1)In Latitude240 ± 498.69 ± 0.564 L + 1205 Cos () = 00.001205 Cos () = + 0.564 L + 258.691452025Cos2()=0.318 L2+291.8 L+ 6692 0.52 (2)For length CDAdd Eq.(1) and Eq.(2)1452025 [Sin2()+Cos2()]=L2 ±2161.63 L+ 2272531.64L2 ± 2161.63 L + 820506.64 = 00.00 (3)Solving Eq. (3) L = 491.455 m.For Bearing of line CD

Substitute in Eq.(1)1205 Sin () = + 0.826 x 491.455 ± 1485.13Sin () = -0.8955

R.B. of CD = N 63º 3422" WW.C.B. of CD = 296º 25 38"

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Example (Mansoura 4/1/2006)

C and D are two stations whose coordinates aregiven below:

Latitude (m)Latitude (m)Departure (m)Departure (m)stationstation

+835+835+380+380CC+1350.50+1350.50--680680DD

From station C is run a line CB of 220 m lengthwith a bearing of 130º. From B is run a line BA of length 640 m and parallel to CD . Find the lengthand bearing of AD?

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Solution:

o

130º

C (380,835)

D (-680,1350.50)

B

A

640 m

N

220 m

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DC = {(1060)2 + (515.5)2 }0.5 = 1178.703 m

W.C.B of DC = tan-11060/515.5 = 115º 56 05

W.C.B of BA = 295º 56 05

LatitudeLatitudeDepartureDepartureW.C.BW.C.BLengthLength(m)(m)

LineLine

--515.50515.50+1060+1060115115ºº 56560505

1178.7031178.703DCDC

--141.413141.413+168.530+168.530130130ºº220220CBCB

+279.902+279.902--575.574575.574295295ºº 56560505

640640BABA

L cosL cos L sinL sin LLADADLL sinsin == -- 652652..983983 mm ,, LL coscos == ++ 377377..011011 mm

(( WW..CC..BB of of AD AD )) == 325325ºº 5959 5757 ,, LengthLength of of AD AD == 754754..005005 mm

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