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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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 2
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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 3
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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 5
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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 6
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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 7
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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 8
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)
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 9
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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 10
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.
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 11
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.
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 12
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)
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 13
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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 14
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’’
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 16
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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 17
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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 18
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
Surveying I Traverse Survey
AAU, Department of Civil Engineering, 2009 19
....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.