CHAPTER 4 - · PDF fileSurveying I Traverse Survey AAU, Department of

Surveying I Traverse Survey

AAU, Department of Civil Engineering, 2009 1

CHAPTER 4

Traverse Survey

4.1 Introduction

A traverse consists of a series of straight lines connecting successive points. The points defining

the ends of the traverse lines are called traverse stations or traverse points. Distance between

successive stations can be measured directly with a tape or indirectly with Stadia or EDM.

Angular measurements or change in direction of lines are observed by a transit or theodolite.

A traverse party is composed of instrument operator; two tape men and a recorder.

Equipments used: - theodolite or transit, leveling rod, steel tape, ranging poles, EDM, and

supporting equipment, plumb bob, stakes and hubs, tacks, axe or hammer, taping pins, notebook,

marking crayon, paint, walkie-talkies, special nails, chisels etc.

Traverse is a convenient and rapid method of establishing horizontal control. It is particularly

useful in densely built up areas and heavily forested regions where lengths of sight are short and

triangulation and trilateration cannot be run comfortably.

Traverses are used for two general purposes

1. For surveying details: -the traverse work provides a system of control points which can

be plotted accurately on the map. Positions of natural and artificial features are located on

the ground relative to the network and these details are plotted on the map by referencing

to the traverse lines and stations.

2. For setting out: - positions of roads, buildings, property lines, and other new

constructions can be established by referencing to a network of traverse lines. The

surveyor can then set out in order to locate the actual position on the ground.

(Hub-traverse station marking peg driven flush with the ground with a tack driven in its top to

make the exact point of reference)

(Stake- extends above ground and driven slopping so that its tip is over the hub. It carries the

number and letter of the traverse station over which it stands.)

4.2 Types of traverse

There are two general classes of traverse: the open traverse and the closed traverse. An open

traverse originates at a point of known position and closes at another point whose location is not

known. No computational checks are possible to check the quality of the work. To minimize

errors distances are observed twice, angles are observed by repetition, magnetic bearings are



observed for all lines and astronomic observations are made periodically. This type of traverse is

applied in mine surveys.

A closed traverse originates on a point of known position and closes on the same point (closed

loop traverse) or on another point of known position (closed link traverse) computational checks

can be applied to a closed traverse to check the quality of the survey.

Fig 1 (a) open traverse (b) closed link traverse (c) closed loop traverse

Traverses can be categorized as interior angle traverse, deflection angle traverse, angle to the

right traverse, azimuth traverse , compass traverse, stadia traverse, or plane table traverse based

on the method used in laying out the traverse lines.

1. Deflection angle traverse

This method of running traverses is widely employed than the other especially on open traverses.

It is mostly common in location of routes, canals, roads, highways, pipe lines, etc.

Azimuth of line =Azimuth of preceding line+ R

Azimuth of line = Azimuth of preceding line- L

In the above figure the azimuth of line AX and DY and are used to check the angular closure for the

traverse.

AXA+2+4-1-3-360=ADY

Where: Al=Azimuth of starting station

A2=Azimuth of closing station

R= deflection to the right

L = deflection to the left

X

A

B

C

D

Y 1

2

3

4



The angular error of closure can be computed and the adjustment of the observed angles is made

assuming equal degrees of precision in observation, the error of closure may be distributed

equally among the deflection angles.

2. Interior Angle Traverse

Interior angle traverse is the one that is employed for closed loop traverse. Successive stations

occupied and back sight is taken to the preceding station. The instrument is then turned on its

upper motion until the next station is bisected/ sighted and the interior angle is observed. The

horizontal circle reading gives the interior angle in the clockwise direction. Horizontal distances

are determined by stadia and angles should be observed twice by double centering.

Azimuth of a line =back azimuth of preceding line + Clockwise interior angle.

Angular closure Check

For closed loop traverse,

n- is the number of stations or sides of a polygon

3. Compass traverse

When compasses are used to run traverses, forward and back beatings are observed from each

traverse station and distances are taped. If local attraction exists at any traverse station, both the

forward and back bearings are affected equally. Thus interior angles computed from forward and

back bearings are independent of local attraction. Since these angles are independent of local

attraction, the sum of these interior angles provides a legitimate indication of the angular error in

the traverse.

Assuming that all bearings are of equal precision and non-correlated, this error is distributed

equally among the number of interior angles. Since none of the traverse lines has an absolute

direction that is known to be correct, it is necessary to select a line affected least by local

attraction.

A

1

2

3

4



4. Angle to the right traverse

This method can be used in open, closed, or dosed loop traverses. Successive theodolite stations

are occupied and back sight is taken to the preceding station with the A vernier set zero. Then

foresight is taken on the next station using the upper motion in the clockwise direction. The

reading gives the angle to the right at the station and angles should be observed by double

sighting.

Azimuth of a line= Back azimuth of preceding line + angle to the right

Generally, the condition of closure can be expressed by

A1+1+2+3+…+n - (n-1)180-A2=0

Where: A1 and A2 are Azimuths of the starting and ending lines

n=number of traverse stations (exclusive of fixed stations)

Any misclosure can be distributed equally to all angles assuming equal precision.

5. Azimuth traverse

This method is used extensively on topographic and other surveys where a large number of

details are located by angular and linear measurements from the traverse stations. Successive

stations are occupied, beginning with the line of known or assumed azimuth. At each station the

theodolite is oriented by setting the horizontal circle index to read the back azimuth (forward

azimuth ± 1800) of the preceding line, and then back sighting to the preceding traverse Station.

The instrument is then turned on the upper motion, and a foresight on the following traverse

station is taken. The reading indicated by the A vernier on the clockwise circle is the azimuth of

the forward line.

Any angular error of closure of a traverse becomes evident by the difference between initial and

final observations Taken along the first line.

6. Stadia traverse

In stadia traverse the horizontal distance between traverse stations is determined by stadia

method. The stadia traverse is sufficiently accurate and considerably more rapid and economical

than corresponding surveys made with theodolite and tape. Its advantage is that elevations can be

determined concurrently with horizontal position.

X

1

2

3

Y

4 Error of closure

Ax1+1+2+3+4 - (4-1)180=A4Y



L1= distance BC observed at station B

L2= distance BC observed at station C

So,

4.3 Traverse computations

Field operation for traverses yields angles or directions and distance for a set of lines connecting

a series of traverse stations. The angles can be checked for angular misclosure and then corrected

so that preliminary adjusted azimuths and bearings are computed. Measured distances have to be

corrected for systematic errors. The preliminary directions and reduced distances are then used in

traverse computations which are performed in plane rectangular coordinate system.

4.3.1 Computation with plane coordinates (latitude and departure)

By considering the figure below

Xk

djk

Yjk

A B

C

Yij

Aj

Ai

j

i k

Y

X

Xj

Yj

Xi

Yi

Xk

Yk

Xij

Xjk

dij



Let the reduced horizontal distance of traverse lines ij and jk be dij and djk respectively, and Ai

and Aj be the azimuths of ij and jk. Let Xij and Yij be the departure and latitude.

Xij=dij sinAi = departure

Yij=dij cosAi = latitude

If the coordinates of i are Xi and Yi, therefore the coordinated of j are:

Xj= Xi+Xij; Yj=Yi+Yij

Xk=Xj+Xjk; Yk=Yj-Yjk

=Xi=Xij+Xik; =Yi+Yij-Yjk

Xjk=djk sinAj; Yjk=djk cosAj

Note: the signs of azimuth functions

If the coordinates for the two ends of a traverse lines are given, the distance between the two

ends can be determined as:

Dij= [(Xj-Xi) 2+ (Yj-Yi)

2]

1/2

The azimuth of line ij from north and south is

After coordinates for all the traverse points (all the departure and latitudes) for all lines have

been computed, a check is necessary on the accuracy of the observations and the validity of

calculations. In a link traverse, the algebraic sum of the departures should equal the difference

between the x coordinates at the beginning and ending stations of the traverse. Similarly, the



algebraic sum of the latitudes should equal the difference between the y coordinates at the

beginning and ending stations.

In a closed loop traverse, the algebraic sum of the latitudes and the algebraic sum of the

departures each must equal zero.

For a traverse containing n stations starting at i=1 and ending at station i=n, the foregoing

conditions can be expressed as follows:

departuresxxxn

i

iin

1

1,1

latitudesyxxn

i

iin

1

1,1

The amount by which the above equations will fail to be satisfied is called simply closure. The

closure correction in departure and latitude are dx and dy, which are of opposite signs to errors,

are:

n

i

iinx XXXd1

1,1)(

n

i

iiny YYYd1

1,1)(

For a closed loop traverse

departuresd x

latitudesd y

4.3.2 Computation with rectangular coordinates

The use of rectangular coordinates permits applications of the principles of analytical geometry

to solve surveying problems.

For a given traverse line ij

Length of line ij,

(xj – xi)

The equation of the line ij for which the coordinates of end stations are given:

i

j (x, y)

Aij



But,

y-yi=cot Aij(x-xi)

Reducing the equation to the form

Ax+By+C=0

Slope of equation

The perpendicular distance between a given coordinate and a line can be determined by using the

equation as:

4.4 Traverse Adjustment

Traverse adjustment is performed to provide a mathematically closed figure that yields the best

estimate for horizontal positions of all the traverse stations.

Error of closure and precision

Error of closure for a given closed traverse is calculated by

22

DLclossure EEE

Where

n

i

L LiLatitudeinErrorE1

n

i

D DiDepartureinErrorE1

and the precision of the measurement law be obtained by

i (x, y)

J (x, y)

p (x, y)

d

Latitude AB

=AB cos (bearing)

Bearing

A

B

Departure AB=AB sin (bearing)



pricisiontheindicateworkFieldPerimeter

Eecision closure Pr

UrbanurbanSubrural

00,100

1,7500

1,

50001precision access. Min.

Example 1

EL= -0.081 and ED=-0.164

183.0)164.0()081.0( 22 closure

51001

51381

46.939

183.0Pr sayecision

Traverse adjustment by the compass rule

The compass rule is used in many survey computations to distribute the errors in latitudes and

departures. The compass rule distributes the errors in latitude and departure for each traverse

course in the same proportion as the course distance is to the traverse perimeter.

Generally, for any traverse station i, let

ii xtocorrectionx

traverse)closedi.e.for traverse,of (perimeter traverseoflength totalL

istation origin to from distance partialL

coordinate-y in the traverse theof correction closure total

coordinate- xin the traverse theof correction closure total

i

t

t

ii

dy

dx

ytocorrectiony

Then the corrections are,

Side Bearing Length Latitude Departure

+N -S +E -W

AB S 6o 15' W 189.53 - 188.404 - 20.634

BC S 29o 38' W 175.18 - 152.268 86.617 -

CD N 81o18' W 197.18 29.916 - - 195.504

DE N 42o59' W 142.39 139.068 - - 30.576

EA N 42o 59' E 234.58 171.607 - 159.933 -

939.46 +340.591 -340.672 +246.550 -246.714



ti

iti

i dyL

Lydx

L

Lx

and

Correction may be applied to departures and latitudes prior to computing coordinates. In this

case,

t

ij

ijt

ij

ij dyL

dydx

L

dx

and

Where ijx and ijy are corrections to departures and latitudes of line ij. Finally, the distances

and directions of the traverse line have to be computed from adjusted coordinate system or

adjusted departures and latitudes.

For example 1 the correction for the latitude of side AB

L

d

dx

x AB

t

AB

46.939

)53.189)(081.0(,

ABAB xLatinCorrection

017.0

Note that if the sign of the error is +, the correction will be minus and vice versa.

Side Latitude

correction

Departure

correction

Balanced latitudes and departure

N S E W N S E W

AB - +0.017 - +0.033 - 188.387 - 20.601

BC - +0.015 +0.030 - - 152.253 86.647 -

CD +0.017 - - +0.035 29.933 - - 195.469

DE +0.012 - - +0.035 139.080 - - 30.551

CA +0.020 - +0.041 - 171.627 - 159.974 -

340.640 340.640 246.621 246.621

4.5 Computation of Area

One of the preliminary objectives of land surveying is to determine the area of the tract. A closed

traverse is run along the boundary of the area. Where the boundaries are irregular and curved, or

where they are occupied by objects that hinder direct measurement, they are located w.r. t. the

traverse line by appropriate angular and linear measurements. The lengths and bearings of all

straight boundaries are determined directly or by computation, the irregular boundaries w.r.t. the

traverse lines by offsets taken at appropriate intervals, the radii and central angles of circular

boundaries are obtained.



The area of land in plane surveying is taken as the projection of the land on a horizontal plane,

and it is not the actual area of the surface of the land.

Methods of determining areas

1. Area from map or plan – by using planimeters

2. Area by triangles

3. Area by coordinates

4. Area by double meridian distances and latitudes (DMD)

5. Areas by ordinate, trapezoid rule and Simpson’s role for irregular boundaries

4.5.1 Area by triangles

The land is divided into a network of triangles.

Given two sides, a and b, and the included angle ^

C between the two sides, the area of

individual triangles is given by

^

sin2

1 CbaA

When the lengths a, b, c of the three sides are given

)(

21 where

))()((

cbas

csbsassA

4.5.2 Area by Coordinates

When the points defining the courses of a tract are coordinated, these coordinates can be used to

calculate the area of the tract. The process involves computing the areas of trapezoids formed by

projecting the sides of the tract up on one of a pair of coordinate areas, usually a true meridian

and a parallel at right angle to this.

Considering the figure below, (a) shows a closed traverse 1, 2, 3, 4 with appropriate x and y

coordinate distances. (b) illustrates the technique used to compute the traverse area.



With reference to (b), it can be seen that the desired area of the traverse is, in effect, area 2 minus

area 1. Area 2 is the sum of the areas of trapezoids 4'433' and 3'322'. Area 1 is the sum of

trapezoids 4'411' and 1'122'.

Expand this expression and collect the remaining terms:

Generally, the above expression can be written as:

2

11

11

1

11

Area

xn

yn

ii

yi

xyn

xn

ii

yi

x

(a)



Where n is the total number of stations in the traverse.

4.5.3 Area by double meridian distances and latitudes (DMD)

When the traverse stations are not coordinated, the area of the tract can be calculated from

adjusted departures and latitudes using the double meridian distance (DMD) method.

The meridian distance of a line is the distance (parallel to the east-west direction) from the

midpoint of the line to the reference meridian. A reference meridian is assumed to pass through

some corner of the tract, usually the most westerly corner for convenience. The double meridian

distance of a line is the sum of the meridian distance of its two extremities.

Then the area is calculated by

2

.1

n

i

Bii LDMD

A

Where latidudecorrectedorbalancdL iB

The DMD procedure is outlined as follows:

1. From a sketch or plot of the traverse, determine which the most western traverse station is.

Consider the reference line for DMDs to be the meridian through that point. The first course

begins at that station, and the succeeding courses follow around the loop in ccw direction

2. Set up a table of the corrected latitudes and departures for each course of the traverse

beginning with the first course as determined in step 1. Include the appropriate sign for each

latitude and departure.

3. Compute the DMD of each course in the table beginning with the first course:

a) The DMD of the first course in equal to the departure of that same course

b) The DMD of any other course is equal to the DMD of the previous course, plus the

departure of the previous course plus the departure of the course itself.

c) The DMD of the last course should equal the departure of that course with its sign

changed.

4. Compute the double areas by taking the product of the latitude and the DMD for each

course. Compute the sum of the double areas with due regard for algebraic sign.

A

B

C

E

D Reference meridian

Meridian distance, MDi



5. Drive the sum by 2 to obtain the enclosed area.

Example 2

Course latitude Departure DMD Double Area

850.7289.10889.10802.669 CD

83.960

16496061.1178

83.96083.96096.139

89.108

DE

63.201

772200,107.2341

63.20163.20100.775 EA

18.680

22727052.1862

18.68018.68002.122

AB

86012017.591

17.59117.59100.174

BC

Double area 980409

Area 299070429801409 uABCDE

Example 3

Example: The following traverse is run to determine the area of a tract. Adjust the traverse and

compute the area of the tract by the coordinate and DMD methods.

Take XA = 520,484.183 m and YA = 424,323.640 m.

'0'0

'0'0

'0'0

'0'0

'0'0

'0'0

3014673.1523025 SFA

1512360.1070059 SEF

3080119.4821564 NDE

4514995.5851535 NCD

1511180.2843065 NBC

4510889.7334545 SAB

anglesInterior length(m)BearingCourse

F

E

D

C

B

A

W

W

W

E

E

E

Solution:

Step 1: Check for angular errors of closure by computing interior angles.

A

B

C

D

E

206035’15’’

64021’15’’



Theoretically, 0

1

720180*)26(180*)2(

nn

i

i

For the given traverse 0720 i hence, no closure error.

(N.B. Distribute the angular error of closure to all angles and adjust the bearings of the lines if

there is any closure.)

Step 2: Compute the departures and latitudes of all lines from preliminary adjusted distances and

directions of the traverse sides and compute the coordinates of the traverse stations (columns 4, 5

and 6)

iij

iiij

i

i

YY coordinate -y

ij line ofAzimuth A XX coordinate -x

cosY Latitude

sinX Departure

y

x

Ad

Ad

ii

ii

coordinateycoordinatexLatitudeDepartureBearingceDis tanStation

Step 3: Calculate the error of closure in departure and latitude and distribute this error using the

compass rule. Adjust the departures and latitudes of the lines. Then calculate the adjusted

coordinates of the traverse stations (columns 7, 8 and 9).

Step 4: From adjusted coordinates, re-establish the distances and directions of all traverse sides.

signs) theof care (take tan 1

ij

ij

Nijyy

xxA

22

ijijij yyxxd



A

WSF

WSE

WND

ENC

ENB

ESA

bearingmcedisStation

'''0

'''0

'''0

'''0

'''0

'''0

403025731.73

404558512.60

204264506.116

001935834.94

004765786.79

001945947.89

)(tan

Check '''0 0000720

F

Aii



1 2 3 4 5 6 7 8 9 Station Distance bearing departure latitudes Coordinates Corrections Corrected Corrected coordinates

X - Y - dep. lat. dep. lat. X - Y -

A 520484.183 424323.640 520484.183 424323.640

89.733 64.276 -62.615 -0.323 -0.634 63.953 -

63.249

B 520548.459 424621.025 520548.136 424620.391

80.284 73.055 33.293 -0.289 -0.567 72.766 32.726

C 520621.514 424294.318 520620.902 424293.117

95.585 55.166 78.059 -0.344 -0.676 54.822 77.383

D 520676.680 424372.377 520675.724 424370.500

119.482 -107.617 51.908 -0.430 -0.844 -108.047 51.064

E 520569.063 424424.285 520567.677 424421.564

60.107 -51.522 -30.957 -0.216 -0.425 -51.738 -

31.382

F 520517.541 424393.328 520515.939 424390.182

73.152 -31.493 -66.026 -0.263 -0.516 -31.756 -

66.542

A 520486.048 424327.302 520484.183 424323.640

S 518.343 0 1.865 3.662 (520484.183) (424323.640) -1.865 -3.662 0 0



t

ij

t

ij

e

dyL

dy

dxL

d

md

ij

ij

tt

22

x

ij line of lat. & dep. toCorrection

662.3dY 1.865dX

sCorrection Closure

1261

518.343

4.110Precision

110.4)662.3()865.1(

4.5.4 Trapezoidal rule

Where the offsets are fairly close together, an assumption that the boundary is straight between

those offsets is satisfactory and the trapezoidal rule may be applied.

1...

232

1 hnhhhh

dA n

4.5.5 Simpson’s one third rule

When the Boundaries are curved, Simpson's one-third rule is a better approach for estimating

areas.

h2 h3

h4 h1

d d d

d d d d d

h1 h2 h3 h4 h4 h5



....2

)2(3

2

22

2)2(

3

2

22 53

45332

231

hhhd

hhd

hhhd

hhdA

1422531 ...4...23

nnn hhhhhhhhd

A

The rule is applicable to areas that have an add number of offsets. If there is an even number of

offsets, the area of all but the part between the last two offsets may be determined with the rule.

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CHAPTER 4 - · PDF fileSurveying I Traverse Survey AAU, Department of