Elements of Land Surveying Lecture Notes
Syllabus:
(i)
Introduction to Surveying – definition of surveying, basic surveying measurements
and deliverables, classes of surveys
(ii)
Distance measurements – direct and indirect distance measurements-taping, optical
(tacheometric, substance bar), EDM;
(iii)
Angular measurements;
(iv)
Traverse
computations
(reduction
of
forward
bearings,
L-D
computation,
computation of coordinates), and adjustments;
(v)
Area computations and subdivision of plots;
(vi)
Spirit and trigonometrical levelling;
(vii)
Introduction to triangulation, trilateration, resection, intersection and radiation as
methods for provision of controls.
Reference:
1) Wolf, P. R. and Brinker, R. C., 1994, Elementary Surveying (9th Ed.), HarperCollins College
Publishers, U.S.A., ISBN 0-06-500399-3;
2) Moffitt F. H. and Bouchard H., 1992, Surveying (9th Ed.), HarperCollins College Publishers,
U.S.A., ISBN 0-06-500059-5;
1
3) Bannister, A. and Raymond S., 1992, Surveying, Longman Group UK Ltd, ISBN 0-582274532;
4) Uren, J. and Price, W. F., 1994, Surveying for Engineers (2nd Ed.), Macmillan Press Ltd.,
London, UK, ISBN 0-333-37081-3
Lecturer: Surv. Daniel Adomako Agyemfrah
MSc. Geodetic Eng., BSc Geodetic Eng., (KNUST), BSc. Admin (Univ. of Ghana)
Ghana School of Surveying and Mapping
Lands Commission, Accra
1 Introduction to Surveying
1.1 Definition of Surveying
Surveying may be defined as the science, art, and technology of determining the relative
positions of natural and man-made features above, on, or beneath the earth’s surface and the
representation of this information either graphically or numerically.
To majority of Engineers, surveying is the process of measuring distances, height differences
and angles on site either for the preparation of large-scale plans or in order that engineering
works can be located in their correct positions on the ground.
Surveying, in a more general sense, can be regarded as that discipline which encompasses all
methods for measuring, processing, and disseminating information about the physical earth
and our environment. Surveying practice therefore involves:
2
(i)
Determination of the shape of the earth and measurement of all facts needed to
determine the size, position, shape, and contour of any part of the earth’s surface,
and the provisions of plans, maps, files and charts recording these facts;
(ii)
Positioning of objects in space, and positioning of physical features, structures, and
engineering works on, above, or below the surface of the earth;
(iii)
Determination of the positions of boundaries of public or private land, including
national and international boundaries, and the registration of those lands with
appropriate authorities;
(iv)
Design, establishment, and administration of land and geographic information
systems, collection and the storage of data within those systems, and analysis and
manipulation of that data to produce maps, files, charts and reports for use in the
planning and design processes;
(v)
Planning of the use, development, and re-development of property, and
management of that property, whether urban or rural, and whether land or
buildings, including determination of values, estimation of costs, and the economic
application of resources such as money, labour, and materials taking into account
relevant legal, economic, environmental and social factors;
(vi)
Study of the natural and social environment, measurement of land and marine
resources, and the use of this data in planning and development in urban, rural and
regional areas.
1.2 Basic Surveying Field Measurements and Deliverables
Basically, field operations in surveying involve measuring distances, height differences and
angles, using ground-based or space-based instruments and techniques. The measured
quantities are processed:
(i)
to determine horizontal positions of arbitrary points on the earth’s surface;
(ii)
to determine elevations or heights of arbitrary points above or below a reference
datum, such as mean sea level;
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(iii)
to determine the configuration of the ground;
(iv)
to determine the lengths and directions of lines;
(v)
to determine the areas of tracts bounded by given lines.
To transfer designed drawings from paper onto the ground, distances, angles and grade lines
are set-out (or laid off) to locate construction lines for buildings, bridges, highways and other
engineering works, and to establish the positions of boundary lines on the ground.
1.3 Geodetic and Plane Surveys
With respect to the assumptions on which the survey computations are based as well as the
orders of accuracies required, surveying may be divided principally into Plane and Geodetic
Surveying. In geodetic surveying, the curved surface of the earth is considered by performing
the computations on an ellipsoid (a curved mathematical figure used to approximate the size
and shape of the earth).
Geodetic methods are employed to determine relative positions of widely spaced monuments
and to compute lengths and directions of the long lines between them. These monuments
serve as the basis for referencing other subordinate surveys of lesser extent. All height
measurements in geodetic surveys are referenced to the surface of the ellipsoid, and are
termed ellipsoidal or geodetic heights.
In plane surveying, relatively small areas of the earth are involved and the surface of the earth
is considered to be a horizontal plane or flat surface. The direction of a plumb line (and thus
gravity) is considered parallel throughout the survey region, and all measured angles are
presumed to be plane angles. All height measurements are referenced to mean sea Level or
the geoid, and are termed orthometric heights.
Field measurements for geodetic surveys are usually performed to a higher order of accuracy
(using special precise instruments and rigorous procedures) than those for plane surveys.
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1.4 Classes of Surveys
The classes of land surveying are:

Topographic Surveys: These are surveys conducted to determine the configuration of
the ground as well as the location of the natural and man-made features of the earth
including hills, valleys, railways etc.

Cadastral Surveys: These are surveys conducted for legal purposes such as deed plans
showing and defining legal property boundaries and the calculation of area(s) involved.

Hydrographic Surveys: These are surveys conducted to determine the position of the
survey vessel, depth of water and to investigate the nature of the sea bed.

Photogrammetric Surveying: It is the science of making precise measurement and
creating detailed maps from aerial images or photographs.

Mining Surveys: These are surveys executed to establish location and boundaries of
mining claims. It also involves the establishment of underground workings horizontally,
vertically and lay out shaft connections.

Engineering Surveying: Surveys executed to locate or lay out engineering or building
works such as roads, railways, tunnels, dams etc.
Questions
1. Distinguish between plane and geodetic surveying.
2. List and discuss four main classes of land surveying.
5
2 Distance Measurements
Measurement of distance between two points on the surface of the earth is one of the basic
operations in surveying. Distances can be measured and set out either directly using tapes or
indirectly using optical theodolites through tachometric techniques or by using electronic
distance meters (EDMs) or Total Stations.
2.1 Types of Distance Measurement
Depending on the relative positions and elevations of the two points involved, the measured
distance could be a horizontal distance, slope distance or vertical distance.
In a 3-D coordinate system,
(i)
a horizontal distance is obtained if the two points have the same Z-value;
(ii)
a slope distance is obtained if the two points have different values for all X, Y, and Z
coordinates;
(iii)
a vertical distance is obtained if the two points have the same X and Y coordinates
but different Z coordinate.
In plane surveying, the distance between two points at different elevations is reduced to its
equivalent horizontal distance either by the procedure used to make the measurement or by
computing the horizontal distance from a measured slope distance. Horizontal and vertical
distances are used in survey drawings, setting out plans, and engineering design works. Slope
distances and vertical distances are used on site during the setting out of designed points.
6
C
S
l
V
e
θ
A
H
o
B
F
i
Note: Distances are corrected for mean sea level and local scale factor corrections only when
the survey is based on the National Grid System.
2.2 Taping: Direct Distance Measurement
Taping is a direct means of determining the straight-line distance between two points using a
tape. The tape may be made of steel, fiberglass or plastic, and may be of length 20 m, 50 m or
100 m. Taping is performed in six steps:
(i)
lining in (through ranging);
(ii)
applying tension;
(iii)
plumbing;
(iv)
marking tape length;
(v)
reading the tape; and
7
(vi)
recording the distance.
When the length to be measured is less than that of the tape, measurements are carried out by
unwinding and laying the tape along the straight line between the points. The zero of the tape
(or some convenient graduation) is held against one point, the tape is straightened, pulled taut
and the distance read directly on the tape at the other point.
2.2.1 Ranging
When the length of a line between the two points exceeds that of a tape, some form of
alignment is necessary to ensure that the tape is positioned along the straight line required.
This is known as ranging and is achieved using ranging poles (or rods) and marking pins (or
arrows). Ranging a line between two points A and B requires two people, identified as the
leader (or surveyor) and the follower (or assistant), and the procedure is as follows:
(i)
Ranging poles are erected as vertical as possible at the points A and B and, for a
measure in the direction of A to B, the zero point of the tape is set against A by the
follower;
(ii)
The leader, carrying a third ranging pole, unwinds the tape and walks towards point
B, stopping just short of a tape length, at which point the ranging pole is held
vertical;
(iii)
The follower steps a few paces behind the ranging pole at point A, and using hand
signals, lines up the ranging pole held by the leader with bottom part of the ranging
pole at A and with the pole at B. This lining-in should be done by the follower
sighting as low as possible on the poles;
(iv)
The tape is now straightened and laid against the pole held by the leader, pulled
taut and the tape length marked by placing an arrow on line;
(v)
For the next tape length, the leader and the follower move ahead simultaneously
with the tape unwound, the procedure being repeated but with the follower now at
the first marking arrow;
(vi)
As measurements proceeds, the follower picks up each arrow and, on completion,
the number of arrows held by the follower indicates the number of whole tape
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lengths measured. This number of tape lengths plus the section at the end less than
a tape length gives the total length of the line.
2.2.2 Step-Chaining: Horizontal Distance Measurements on Sloping Ground
Step-chaining is a field procedure for directly obtaining horizontal distance between two points
on sloping ground without using angle-measuring or levelling instruments. Two men, a
surveyor (leader) and an assistant (follower), are required, and with reference to Fig. 2.2, the
procedure is as follows:
9
A
D
D
i
1
D
2
D
B
3
D
F
i
(i)
To measure D1, the zero end of the tape is held at A and the tape then held
horizontally and on line towards B against a previously lined-in ranging pole;
10
(ii)
At some convenient tape graduation (preferably a whole metre mark), the horizontal
distance is transferred to ground level using a plumb line (i. e. a string line with a
weight attached), a marking arrow or a ranging pole;
(iii)
The leader notes the length of the first step in his book, and the tape is now moved
forward and the process is repeated to measure D2 and D3 in a similar manner; and
(iv)
The sum of the steps D1, D2 and D3 gives the required horizontal distance between A
and B.
The length of steps which can be adopted is limited by the gradient. At no time should the tape
be above the surveyor’s eye level, because plumbing becomes very difficult. As the gradient
increases the length of step must therefore decrease.
2.2.3 Slope Measurements
In measuring the horizontal distance between two points on a steep slope, rather than break
tape every few meters, it may be desirable to tape along the slope and compute the horizontal
distance. This requires measurement also of either the angle of inclination (θ) or the difference
in elevation (d) as indicated in Fig. 2.1. The slope angle can be measured using a hand-held
device called an Abney Level (see Fig. 2.3) but where better accuracy is required, a theodolite
is used to measure the slope angle.
Fig 2.3 Abney level
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To use an Abney level, an observer first distinctly marks his eye height (h in Fig. 2.4) on a
ranging pole which is then placed at point B. Standing at point A and looking down the sighting
tube, the cross-wire is seen and is set against the mark on the ranging pole at B. The
observer’s line of sight will be A'B', which is parallel to AB.
Fig 2.4 Measuring slope angle with Abney level
To record the slope angle θ, the milled wheel is turned until the image of the bubble appears
centrally against the cross-wire when viewed through the sighting tube. A fine adjustment is
provided by the slow motion screw. A simple vernier, attached to the milled wheel, is then read
with the aid of a small reading glass against the scale attached to the sighting tube. This gives
a measure of θ to within 10 minutes of arc.
Worked Example 1:
Calculate the plan length for a measurement of 126.300 m along a gradient of 2° 34′.
Solution 1:
Let θ be the inclination (or slope) angle = 2º 34'
Plan (or Horizontal length) = slope length X cos θ
= 126.300 X cos 2º 34'
= 126.173 m
Worked Example 2:
Calculate the plan length where a distance has been measured along a slope of 1 in 3 and
found to be 149.500 m.
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Solution 2:
Let θ be the inclination (or slope) angle
For a slope of 1 in 3, cos  
3
10
Plan length = slope length  cos  = 149.500 m 
3
 141.828 m
10
Questions
1.
A horizontal distance of 745.000 m is to be established along a line that slopes at a
vertical angle of 5º 10'. What slope distance should be measured off?
2.
A distance of 3236.86 ft was measured along a smooth slope. The slope angle was
measured and found to be 3º 22'. What is the horizontal distance?
2.2.4 Reduction of Slope Measurements by Difference in Elevation
Measurements made on the slope (L) can be reduced to their corresponding horizontal
distances (H) using Pythagoras theorem if the differences in elevation between the two ends of
the tape (d) have been measured by levelling.
H
L  d   L  d 
2
2
2
2
1
2
Worked Example 3:
A distance of 290.430 m was measured along a smooth slope from A to B. The elevations of A
and B were measured and found to be 865.2 and 891.4 m, respectively. What is the horizontal
distance from A to B?
Solution 3:
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Slope distance, L = 290.430 m
Elevation difference, d = 891.4 – 865.2 = 26.2 m
From Pythagoras theorem,
Horizontal distance, H 
L  d  
2
2
(290.43)2  (26.2)2  289.25 m
Question
A line measures 1446.25 m along a constant slope. The difference in elevation between the
two ends of the line is 57.24 m. Calculate the horizontal length of the line.
2.2.5 Errors in Making Measurements
An error is the difference between a measured value for a quantity and its true value. That is,
E  M  T , where E is the error in a measurement, M the measured value, and T its true
value.
It can be unconditionally stated that:
(i)
no measurement is exact;
(ii)
every measurement contains errors;
(iii)
the true value of a measurement is never known; and therefore
(iv)
the exact error present is always unknown
Note that mistakes are observer blunders and are usually caused by a misunderstanding of the
problem, carelessness, fatigue, missed communication, or poor judgement. Examples are
transposition of numbers such as recording 73.96 as 79.36; failure to include a full tape length.
Mistakes can be detected by systematic checking of all work, and eliminated by redoing part of
the job or even all of it.
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2.2.6 Sources of Error in Making Measurements
There are basically three main sources of error in measurements namely natural, instrumental
and personal.

Natural sources of error are caused by variations in wind, temperature, humidity,
atmospheric pressure, atmospheric refraction, gravity, and magnetic declination. An
example is a steel tape whose length varies with changes in temperature.

Instrumental Errors are caused by any imperfection in the construction or adjustment of
survey instruments. For example, the graduations on a scale may not be perfectly
spaced, or the scale may be warped. The effect of many instrumental errors can be
reduced, or even eliminated, by adopting proper surveying procedures or applying
computed corrections.

Personal errors arise from the inability of the individual (observer) to make exact
observations due to limitations of the human senses of sight and touch. As an example,
a small error occurs in the measured value of a horizontal angle if the vertical cross-hair
is not aligned perfectly on the target.
2.2.7 Classification of Errors
Errors in measurements are of two classes: Systematic and Random.
2.2.7 (i) Systematic Errors
A systematic error is any biasing effect in the environment, methods of observation or in the
measuring instrument which introduces an error into a measurement such that the measured
value is either too high or too low.

Systematic errors are attributable to known circumstances. They could be due to
instrumental imperfections or effects of the environment on the measurement.

They are usually constant (having the same magnitude and sign) throughout an
operation.
15

Systematic errors which change during a measurement process are termed drifts. Drift
is evident if a measurement of a constant quantity is repeated several times and the
measurements drift one way during the process, for example if each measurement is
higher than the previous measurement which could occur if the instrument becomes
warmer during the measuring process.

They conform to mathematical and physical laws; thus their magnitudes or values could
be computed and appropriate corrections (i.e. “negative the error”) can be applied to
mitigate them.

Cumulative observations will increase or propagate the effect of systematic errors.

Systematic errors can be detected by measuring already known quantities through a
process called calibration or by comparing the measurements with ones made using a
different instrument known to be more accurate.
The principal systematic errors in linear measurements made with a tape are:
(i)
incorrect length of tape;
(ii)
tape not horizontal;
(iii)
fluctuations in the temperature of the tape;
(iv)
incorrect tension or pull;
(v)
sag in the tape;
(vi)
incorrect alignment; and
(vii)
tape not straight
2.2.7 (ii) Random Errors
A random error is the irreproducibility in making repeated/replicate measurements, and it
affects the precision of the measured quantity.
Repeated observation of a constant quantity . Random Errors are associated with the skill and
vigilance of the surveyor. They are introduced into each measurement mainly due to the
16
surveyor’s inability to take the same measurement in exactly the same way to get exactly the
same value. They represent the residual error after all other errors have been eliminated. They
are compensating and generally unavoidable, and usually conform to the law of probability.
Taking the mean of repeated observations minimizes their effects.
The characteristics of Random errors are as follows:
1. Small errors occur more frequently than large ones
2. Positive and negative errors of equal magnitude occur with equal frequency; that is they
are equally probable.
3. Large errors are rare (infrequent) and are more likely to be mistakes or untreated
systematic errors.
2.2.8 Precision and Accuracy
A discrepancy is the difference between two measured values of the same quantity. A small
discrepancy indicates there are probably no mistakes and random errors are small. Small
discrepancies do not preclude the presence of systematic errors, however.
Precision refers to the degree of closeness or consistency of a group of measurements, and is
evaluated on the basis of discrepancy size. If multiple measurements are made of the same
quantity and small discrepancies result, this indicates high precision. The degree of precision
attainable is dependent on equipment sensitivity and observer skill.
Accuracy denotes the absolute nearness of measured quantities to their true values. Since the
true value is seldom known, accuracy is generally indeterminate in practice.
Example
Two groups of students measured a line of 100 m nominal length with a tape, and obtained the
following results:-
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2.2.9 Applying Corrections to Tape Measurements
2.2.9(i) Incorrect Length of Tape: Standardization Correction
An error due to incorrect length of a tape occurs each time the tape is used. If the true length,
determined by standardization or calibration under specific conditions, is not exactly equal to its
nominal value as given by the manufacturer, then the correction to be applied to the measured
length is given by
 A  NL 
CL   L
 ML
 NL 
and
TL  M L  CL
A 
  L ML
 NL 
where CL = correction to be applied to the measured length of a line to obtain the true length
AL = actual tape length (obtained from standardization)
N L = nominal tape length (given by manufacturers)
M L = measured (or recorded) length of line
TL = corrected (or true) length of line
Worked Example 4:
A 100-m steel tape when compared with a standard is actually 100.020 m long. What is the
corrected length of the line measured with this tape and recorded to be 565.750 m?
Solution 4:
Nominal length, N L = 100.000 m
Actual length, AL = 100.020 m
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 100.020  100.000 
 565.75  0.113 m
100.000


Standardization correction, CL  
Corrected length of line, TL  M L  CL  565.750  0.113  565.863 m
Note (i): AL  N L = amount by which a full tape length is too long or too short = 0.020 m
ML
= number of tape lengths in the measured line = 5.6575
NL
CL = 5.6575 x 0.020 m = 0.113 m
Note (ii): In measuring unknown distances with a tape that is too long, a correction must be
added. Conversely, if the tape is too short, the correction will be minus, resulting in
a decrease.
2.2.9(ii) Changes in temperature: Temperature Correction
When the field temperature during the period of making the measurement differs considerably
from the standard temperature of the tape, a temperature correction must be computed and
applied to the measured length. The amount of correction for temperature is determined as
follows:
Ct  L Tm  Ts 
and
LC  L  Ct
where Ct = the temperature correction to be applied (in feet or metres)
L = the length of the tape actually used (in feet or metres)
 = the coefficient of linear expansion of the material of the tape
Tm = the tape temperature at the time of measurement
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Ts = the tape temperature when it has standard length
2.2.9(iii) Tape Not Horizontal: Slope Correction
If the tape is assumed to be horizontal but actually is inclined, an error is introduced. The
amount of this error is computed as follows:
Cs  L 1  cos  

d2
L
and
H  L  Cs
where d  h = the height difference between the ends of the line
L = measured slope distance
 = slope angle
H = horizontal distance
2.3 Indirect Distance Measurements
An indirect measurement is secured when it is not possible to apply a measuring instrument
directly to the quantity to be measured. The required distance is therefore determined by its
relationship to some other measured value or values.
2.3.1 Optical Distance Measuring Techniques
2.3.1 (i) Stadia Tacheometry
Tacheometry is a surveying method used to quickly determine the horizontal distance to, and
elevation of, a point. It is used primarily to provide the position and heights of the spot heights
20
from which contours are interpolated, and for the positions of points of detail during the
production of large and medium-scale (1/500 to 1/5000) topographic surveys with contour
intervals ranging from 0.5 m to 5 m. Tacheometry can also be used to provide horizontal and
vertical control. Instruments required for stadia measurements are:
(i) a theodolite or level with a telescope equipped with two or more horizontal lines,
called stadia hairs or stadia lines. The distance between the upper and lower stadia
hairs is fixed and is called stadia interval;
(ii) a levelling staff; and
(iii) a tape.
Suppose the theodolite is set up at station P and the levelling staff held vertically at station X as
shown in Fig. 2.5.
Fig. 2.5: Stadia Tacheometry
Measurements taken are:
(i)
staff intercept, i.e. the apparent intercepted length between the top and bottom
hairs read on the levelling staff;
(ii)
the vertical angle read from the vertical circle of the theodolite;
(iii)
centre hair reading of the staff at X;
(iv)
height of the theodolite measured using the tape.
Indirectly computed quantities are:
21
(i)
Horizontal distance H between station P and station X, which is given by the
horizontal component of the inclined line of sight, obtained indirectly as follows:
H  Ks cos 2   C cos 
(ii)
The vertical component of the inclined line of sight given by:
V
1
 Ks sin 2   C sin 
2
V is the difference in height from the instrument axis to the centre hair staff
reading. It is positive for angles of elevation and negative for depressions.
(iii)
The reduced level of the staff position is given by
RLX  RLP  hi  V  m
where K = multiplying constant of the telescope, usually set at 100;
s = stadia interval on the staff= upper stadia reading – lower stadia reading;
m =centre hair reading of the staff at X;
 = vertical angle or inclination of line of sight to the horizontal;
hi =height of the theodolite at P;
C = additive constant, usually taken to be zero (0); and
RLP , RLX = reduced level of stations P and X, respectively.
Question:
A theodolite having a multiplying constant of 100 and an additive constant of 0 was
correctly centred and levelled at a height of 1.490 m above a traverse station P of
reduced level 46.870 m during a stadia tacheometric survey. A levelling staff was held
vertically at the bases of two electric poles labelled A and B in turn, and the readings
shown in Table 1 were taken.
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Table 1
Staff position
Staff reading (m)
Vertical circle reading
Horizontal circle reading
Electric pole A
1.981, 1.497, 1.013
90° 08' 20"
169° 33' 45"
Electric pole B
1.773, 1.456, 1.142
88° 13' 00"
275° 12' 25"
Compute:
(i)
the horizontal distances of lines PA and PB; and
(ii)
the reduced levels at the bases of electric poles A and B.
2.3.1 (ii) Subtense Tacheometry
Subtense tacheometry is an indirect distance measuring procedure involving reading the angle
subtended by two precisely spaced targets on a 2-m long subtense bar. The bar is mounted on
a tripod, levelled by means of a level vial, and aligned perpendicular to the survey line by
means of a sighting device on top of the bar.
Fig. 2.6: Subtense Bar
The horizontal angle  between the targets is measured with a theodolite set over the other
end of the line (Fig. 2.7).
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Fig. 2.7: Subtense Bar Tacheometry
The horizontal distance H between the theodolite and the target is computed using the
relation:
b
 2
2
H
 
tan 
H
 
b
cot 
2
2
 2
H  cot 
b=2 m
Note: Distances determined by the subtense method are always horizontal, even though
inclined sights are taken, because
 is measured in a horizontal plane.
2.3.3 Electronic Distance Measuring Instruments (EDMs)
Electronic distance measuring (EDM) instruments determine distances by indirectly measuring
the time it takes electromagnetic energy of known velocity to travel from one end of a line to
the other and return. EDMs have practically replaced the tape for measuring all but relatively
short distances. Now EDM instruments have made it possible to obtain accurate distance
measurements rapidly and easily. Given a line of sight, long or short lengths can be measured
over bodies of water, busy freeways, or terrain that is inaccessible for taping.
24
The first EDM instrument, called the Geodimeter (an acronym for geodetic distance meter) was
introduced in 1948, used visible light and could measure distances up to 40 km. The second
EDM instrument, called the tellurometer, was introduced in 1957, used microwaves and was
capable of measuring distances up to 80 km, day or night. In the current generation, EDM
instruments have been combined with digital theodolites and microprocessors. The resulting
devices, called Total Station instruments, can measure simultaneously and automatically both
distances and angles. The microprocessor receives the measured slope distance and vertical
angle, calculates horizontal and vertical distances, and displays them in real time.
Measuring Principle of EDM Instrument
The instrument sends out a beam of light or high-frequency microwaves from one end of a line
to be measured, and directs it toward the far end of the line. A reflector or transmitter-receiver
at the far end reflects the light or microwaves back to the instrument where they are analyzed
electronically to give the distance between the two points.
25
Fig. 2.8: Measurement principle of EDM
The indirect time measuring scheme of EDM involves determining how many cycles of
electromagnetic energy are required to travel the double path distance. The frequency (time
required for each cycle) is precisely controlled by the EDM instrument and thus known, so the
total travel time becomes known. Multiplying total time by velocity, and dividing by 2, yields the
unknown distance.
Note: Air temperature, atmospheric pressure and relative humidity are atmospheric conditions
that affect the velocity of propagation of light and microwaves, and therefore affect the
accuracy of the measured distance. Knowledge of these conditions allows a determination of
the refractive index of the air, which must be known to compute the velocity of light or
microwaves under given meteorological conditions.
Assignment 1
1.
Tape readings totaled 357.879 m when a measuring tape of 50 m nominal length was
used to measure a line. The tape was calibrated later and found to have actually
measured 50.009 m at the temperature at which the line had been measured. Calculate
the correct length of the line.
2.
3.
The slope length of a line is 123.400 m and the slope is 4º 53'.
(i)
What is the horizontal length of the line?
(ii)
Calculate the error that will be made if the slope is taken as 5º exactly?
A line AB is measured in three sections AX, XY, and YB using a fiberglass tape of nominal
length 20 m. The tape was read to the nearest 0.01 m and distances obtained for the
three sections AX, XY, and YB were 79.45 m, 8.70 m, and 126.35 m, respectively.
26
There is a constant slope from A to X and from Y to B, and stepping was carried out
between X and Y due to very steep ground. The reduced levels of points A, X, Y, and B
are 37.62 m, 32.14 m, 19.47 m, and 20.21 m, respectively. Before measurement, the
tape was measured against a reference steel tape and found to be 20.015 m. Calculate
the horizontal length of AB.
4.
A theodolite having a multiplying constant of 100 and an additive constant of 0 was
correctly centred and levelled at a height of 1.528 m above a traverse station Q of reduced
level 56.875 m during a stadia tacheometric survey. A levelling staff was held vertically at
the bases of two electric poles labelled P and T in turn, and the readings shown in Table 2
were taken.
Table 2
Staff position
Staff reading (m)
Vertical circle reading
Horizontal circle reading
Electric pole P
1.773, 1.456, 1.142
90° 10' 20"
165° 33' 45"
Electric pole T
1.981, 1.497, 1.013
86° 23' 00"
270° 12' 25"
Compute:
(iii)
the horizontal distances of lines QP and QT; and
(iv)
the reduced levels at the bases of electric poles P and T.
27
3 Angular Measurements
Angular measurements are carried out in surveying for orientations of lines and determination
of locations of points. Angles are directly measured in the field using Total Stations,
theodolites, compasses, sextants, or the Abney level. An angle can be measured indirectly by
the tape method (or linear measurements) and its value computed from the relationships of
known quantities in a triangle or other geometric figure. To determine an angle, the following
three basic requirements must be met:
(i)
reference or starting line;
(ii)
direction of turning; and
(iii)
angular distance (i.e. value of the angle).
28
T
o
D
i
S
t
θ
F
i
S
t
29
Angles measured in surveying are classified as horizontal or vertical depending on the plane in
which they are measured. Horizontal angles are measured in the horizontal plane and vertical
angles are measured in the vertical plane.
3.1 Measuring Horizontal Angle
A horizontal angle is the angle formed in a horizontal plane by two lines extending from the
same point. In surveying, a horizontal angle has a direction or sense; it is considered clockwise
if it is measured to the right or anticlockwise if it is measured to the left. Horizontal angles are
the basic measurements needed for determining bearings and azimuths.
Suppose it is desired to measure the horizontal angle at B from A to C, as shown in Fig. 3.2. A
theodolite is set up over point B, centred, and levelled. A backsight is taken to a target set up
at point A and the initial horizontal circle reading is recorded. The telescope is rotated in the
direction of point C and a foresight is taken to a target set up at point C, and the final
horizontal circle reading is recorded. The horizontal angle is obtained by taking the difference
between the initial and final horizontal circle readings.
30
S
t
S
t
B
a
c
5
3
F
o
r
S
t
F
i
3.2 Description of theodolite
Theodolites are precision instruments used extensively in construction work for measuring
angles in the horizontal and vertical planes. Many different theodolites are available for
measuring angles and they are classified according to the smallest reading that can be taken
with the instrument. For example, a 1'' theodolite is one which can be read to 1'' directly
without estimation.
3.3 Setting up the theodolite
The sequence of operations required to get the theodolite ready for angle measurements is as
follows:
31
(i) Set up the tripod over the ground point such that the top is approximately horizontal;
(ii) Mount the theodolite on the tripod;
(iii) Centre the theodolite over the ground mark using the optical plummet or the plumb line;
(iv) Level the theodolite by centralizing the horizontal plate bubble; and
(v) Remove parallax.
3.4 Double sighting with the theodolite
When a theodolite is set up over a survey mark and properly levelled, the position of the
vertical circle with respect to the observer when looking through the telescope is used to define
the two positions or faces of the theodolite:
(i) Face Left (L): vertical circle is to the left of observer;
(ii) Face Right (R): vertical circle to the right of observer.
The theodolite can be swung in two directions:

Swing Left (L) or anticlockwise

Swing Right (R) or clockwise
When a theodolite is in face left position when sighting one of the points, it is usually swung to
the left to sight the other point. Similarly, when it is in face right position, it is swung to the
right to sight the other point. Combining the face and swing we have:
LL meaning face left, swing left; and
RR meaning face right, swing right
Double sighting consists of making a measurement of a horizontal or a vertical angle once with
the telescope in the direct or face left position and once with the telescope in the reversed,
inverted, plunged or face right position. The act of turning the telescope upside down, that is,
32
rotating it about the transverse axis, is called “plunging” or “transiting” the telescope. When the
telescope is in its plunged position, the telescope bubble is on top of the telescope.
3.5 Booking and calculating horizontal angles
The horizontal angle booking for a theodolite is illustrated in Table 3.1.
Table 3 Horizontal Angle Booking
Instrument
Face &
Back
Station
Swing
Station Station Circle Reading
LL
A
LL
Fore
Horizontal
Included
Angle
Mean Included
Angle
90º 12' 30''
C
107º 31' 10''
17º 18' 40''
B
17º 18' 45''
RR
RR
C
A
287º 31' 20''
17º 18' 50''
270º 12' 30''
Question 1:
A horizontal angle is measured at E from D to F by the method of double centering. A backsight
on D gives a circle reading of 0º 00' 40'' and a foresight on F gives a circle reading of 146º 40'
20''. After plunging the telescope, a backsight is made to D and a foresight is made to F. The
final circle reading is 293º 19' 40''. What is the value of the angle?
Question 2:
A backsight is made at K to J, and the horizontal circle reads 312º 14' 30''. A foresight is made
to L, and the circle reads 14º 42' 00''. The telescope is reversed, a backsight is again made to J
and a foresight to L. The final circle reading is 77º 09' 00''. What is the value of the angle?
3.6 Measuring Vertical and Zenith Angles
A vertical angle is an angle measured in a vertical plane from a horizontal line upward or
downward to give a positive or a negative value, respectively. Positive and negative vertical
33
angles are sometimes referred to as elevation or depression angles, respectively. A vertical
angle thus lies between 0 and ± 90º.
A zenith angle is an angle measured in a vertical plane downward from an upward directed
vertical line through the instrument. It is thus between 0º and 180º.
Vertical (or zenith) angles are used in trigonometric levelling, stadia tacheometry, and for
reducing measured slope distances to horizontal.
To measure the vertical angle of the line of sight from one point to another, the theodolite is
set up over the first point, centred, and levelled. The line of sight is brought in the direction of
the other point, and the telescope bubble is centred. The reading of the vertical circle is
recorded as the initial reading. The line of sight is then raised or lowered and is directed
accurately to the second point by means of the vertical clamp and tangent screw. The reading
of the vertical circle is recorded as the final reading. The correct vertical angle is the difference
between the initial and the final readings.
3.8 Measuring Interior Angles
Interior angles are angles formed within a closed figure between the adjacent sides. In Fig.
3.3, there are five interior angles to be measured. They are the angles at L, M, N, O, and P.
34
N
O
1
1
M
9
0
9
7
1
1
1
2
L
P
F
i
If the theodolite is set up at each station in turn in the order just given, all the interior angles
would be measured in an anticlockwise direction. Thus the interior angle at L is measured from
P to M, that at M from L to N, that at N from M to O, that at O from N to P, and that at P from
O to L. On the other hand, if the instrument is set up in turn at L, P, O, N, and M, then all the
interior angles would be measured clockwise.
Note: If the stations are occupied in a clockwise direction, the interior angles are measured in
an anticlockwise direction. On the other hand, if the stations are occupied in an
anticlockwise direction, the interior angles are measured in a clockwise direction.
35
3.8.1: Finding the angular misclosure in a closed plane figure
The sum of the interior angles in a closed plane figure must equal (n – 2) x 180º, in which n is
the number of sides in the figure. This relation furnishes a check on the accuracy of the field
measurements of the angles and a basis for distributing the error.
Table 3.2: Interior angles in a closed figure
Station
L
M
125º 11' 20''
Observed Angle
N
97º 20' 30''
115º 32' 10''
O
90º 56' 30''
P
111º 01' 30''
Sum of measured interior angles = 540º 02' 00''.
Expected sum of interior angles = 540º 00' 00''.
Angular misclosure = measured sum – expected sum = 540º 02' 00'' - 540º 00' 00'' = + 02'
00''
3.8.2: Adjusting observed interior angles
Interior angles are adjusted by distributing the misclosure over all the measured angles as
shown in Table 3.3. For the above, the adjustment per angle will be – (02' 00'')/5, since five (5)
interior angles were measured:
Table 3.3: Adjustment of interior angles
Station
Observed angle
Adjustment Adjusted Interior angle
L
125º 11' 20''
-24''
125º 10' 56''
M
97º 20' 30''
-24''
97º 20' 06''
N
115º 32' 10''
-24''
115º 31' 46''
O
90º 56' 30''
-24''
90º 56' 06''
P
111º 01' 30''
-24''
111º 01' 06''
36
Sum
540º 02' 00''
- 02' 00''
540º 00' 00''
3.9: Meridians (or Reference Lines) for Bearings
3.9.1: Astronomical (or Geographic or True) Meridian
The true meridian at a point on the earth’s surface is defined as a plane passing through the
point and containing the earth’s axis of rotation. In other words, a true meridian is an
imaginary line on the mean surface of the earth joining the geographic north and south poles.
3.9.2: Magnetic Meridian
The magnetic meridian at a point on the earth’s surface is defined as a line passing through the
point and joining the earth’s magnetic north and south poles. A freely suspended needle of a
magnetic compass will always come to rest in a north-south direction defining the direction of
the magnetic meridian.
Unlike the true meridian, whose direction is fixed, the magnetic meridian varies in direction.
The amount and direction by which the magnetic meridian deviates from the true meridian is
called the magnetic declination. Declination may also be defined as the angular difference
between the true north and magnetic north. Declination is positive or plus when the needle of a
magnetic compass points east of true north, and it is negative or minus when the needle points
west of true north.
3.9.3: Assumed Meridian
Assumed meridian is any arbitrary line of a survey chosen as a reference line for convenience
in a survey of limited extent. It is usually taken to be in the general direction of the true
meridian.
3.9.4: Convergence of Meridians
True meridians on the surface of the earth are lines of geographic longitude, and they
converge toward each other as the distance from the equator toward either of the poles
increases. The amount of convergence between two meridians in given vicinity depends on:
37
(i)
its distance north or south of the equator; and
(ii)
the difference between the longitudes of the two meridians.
Magnetic meridians tend to converge at the magnetic poles but the convergence is not regular
and it is not readily obtainable.
3.9.5: Grid Meridians
Grid meridians are lines that are parallel to the central meridian chosen as a reference line in a
map projection system of a nation. In Ghana, the central meridian chosen as a reference line is
longitude 1º west of Greenwich, and all our grid meridians are parallel to it.
3.10 Bearing of a line
The bearing of a line AB is defined as the clockwise horizontal angle measured from the
meridian passing through A to the vertical plane containing the line. Note that the bearing of a
line gives the direction of that line with respect to a reference meridian, and that in plane
surveying,
(i)
bearing is always measured from the north (for countries in the northern
hemisphere like Ghana);
(ii)
meridians passing through other points in the survey are assumed to be parallel
to the meridian passing through the origin of the survey.
Bearings may be of different types depending on the type of reference meridian: magnetic
bearing is with respect to magnetic meridian, true bearing is with respect to true meridian, grid
bearing is with respect to grid meridian, and assumed bearing is with respect to assumed
meridian.
3.10 .1 Whole Circle Bearing & Quadrant Bearing
38
The bearing of a line may be expressed as a whole circle bearing, having a value between 0º
and 360º, or as a quadrant bearing having a value between 0º and 90º and which states
whether the angle is measured from the north or the south and also whether the angle is
measured toward the east or toward the west.
39
N
N
B
W
C
M
W
C
A
P
S
S
F
i
40
The whole circle bearing of line AB shown in Fig. 3.4 is 70º while that of line MP is 240º 20' 30''.
The quadrant bearing of line AB is N 70º E and that of line MP is S 60º 20' 30'' W.
Note:
(i)
If a whole circle bearing (WCB) from north is between 0º and 90º, the line is in
the northeast quadrant and the quadrant bearing is equal to the whole circle
bearing.
(ii)
If the WCB is between 90º and 180º, the line is in the southeast quadrant, and
the quadrant bearing is 180º minus the WCB;
(iii)
If the WCB is between 180º and 270º, the line is in the southwest quadrant, and
the quadrant bearing is the WCB minus 180º;
(iv)
If the WCB is between 270º and 360º, the line is in the northwest quadrant, and
the quadrant bearing is 360º minus the WCB.
Questions
1.
2.
Convert the following quadrant bearings to whole circle bearings:
(a)
S 15º 56' 30'' E
(b)
N 18º 15' 20'' W
(c)
S 85º 16' 35'' W
Convert the following whole circle bearings to quadrant bearings:
(a)
215º 56' 30''
(b)
326º 15' 10''
(c)
172º 38' 50''
41
3.10.2 Back Bearing of a line
The back bearing of a line is the bearing of the line running in the reverse direction. According
to the assumption in plane surveying, the meridians passing through the end points of a line
are parallel, and that the back bearing of a line differs from its forward bearing by 180º. That
is, back bearing of a line = forward bearing of that line ± 180º.
In Fig. 3.4, the back bearing of line AB = the bearing of line BA = 70º + 180º = 250º. Similarly,
the back bearing of line MP = the bearing of line PM = 240º 20' 30'' - 180º = 60º.
Note: The back bearing of a line expressed in the quadrant form can be obtained from the
forward bearing by simply changing the letter N to S and S to N, and also changing E to W or
W to E.
Question
Find the back bearings of the following lines, express your answers in whole circle format:
(i)
S 15º 56' 30'' E
(ii)
N 85º 40' 30'' W
(iii)
S 65º 35' 20'' W
3.11 Determining directions with the magnetic compass
When using a pocket compass, the observer occupies one end of the line whose magnetic
bearing he wishes to obtain. He holds the compass level, releases the needle by lifting the
sights to their sighting position, sights the other end of the line, and allows the needle to come
to rest. He then depresses the needle clamp and reads the circle at the north end of the
compass needle. This reading gives him the magnetic bearing of the line directed from his
position to the point sighted.
To obtain the back magnetic bearing as a check, he occupies the second point and sights back
to the first point, reading the north end of the needle as before. The bearings should show
reasonable agreement.
42
3.12 Local attraction
Local attraction is the interference with the earth’s magnetic field caused by magnetic materials
(such as electric cables, small masses of iron or iron ore deposits) present in the vicinity of the
point where a magnetic compass is being used to take magnetic bearings. The net effect of
local attraction is that the needle of the magnetic compass is forced to deflect from the actual
magnetic meridian, and the resultant reading would be in error.
Local attraction is detected when the forward and back magnetic bearings of a line differ by
more than 180º. This effect is minimized by meaning each line’s forward and back bearings,
and eliminated when stations with no local attractions are encountered.
Question
Table below shows bookings of a closed loop compass traverse survey conducted by a student.
Using any of the methods of eliminating the effect of local attraction, obtain the corrected
forward bearings of the traverse legs.
Line
Forward Bearing (FB)
Back Bearing
A–B
32º 00'
213º 30'
B–C
78º 30'
257º 00'
C–D
107º 00'
285º 00'
D–E
120º 00'
302º 00'
E–A
265º 00'
85º 00'
Correction
Corrected FB
4 TRAVERSING
4.1 Definition of traversing
Traversing is the act of establishing a network of points (or survey stations) whose horizontal
coordinates are determined by a combination of distance and angle measurements between
successive lines joining them. A traverse is a series of consecutive lines whose lengths and
43
bearings have been determined from field measurements. The lengths are horizontal distances
and the bearings are either true, magnetic, assumed, or grid bearings.
4.2 Importance of traversing
Traverse surveys are made for many purposes including:
(i)
To determine the positions of existing boundary markers or pillars;
(ii)
To establish the positions of boundary lines;
(iii)
To determine the area encompassed within the confines of a boundary;
(iv)
To establish control for locating railroads, highways, and other construction
works;
(v)
To
establish
controls
for
topographic,
cadastral,
hydrographic,
and
photogrammetric mapping.
4.3 Types of traverse surveys
There are two basic kinds of traverses: closed and open. Two categories of closed traverses
exist, and these are the polygon and the link traverses. A closed-polygon traverse starts from
one point and closes on that same point, thus forming a closed figure which affords a check on
the internal accuracy of the measured angles.
44
E
A
A
B
B
B
C
D
There are no starting and/or
finishing control points
E
A
D
C
C
D
Point A is both the starting and
finishing point
A, B, D and E are points of
known coordinates, and are
termed Points of Departures
45
Fig 4.1 (i) Polygon Traverse
(ii) Link Traverse
(iii) Open Traverse
A link closed traverse originates at a point of known coordinates and closes on another known
station of equal positional accuracy. Closed traverses provide checks on the measured angles
and distances.
An open traverse is one in which the horizontal coordinates of either the starting point or the
finishing point is unknown or both are unknown. Open traverses do not offer any means of
checking for errors and mistakes.
4.4 Traverse Computations
Figure 4.2 below shows measured angles at traverse stations and lengths of traverse legs for a
closed loop traverse as obtained from an angle book, and it would be used to illustrate traverse
computations.
N
F
3
17
E
7A
3
1
1
1
1
2
4
8
2
1D
9
3B
5
3
16
02
5
C
46
5
2
47
Figure 4.2: Traverse abstract from angle book
Step 1: Finding the angular misclosure
The angular misclosure for an interior-angle traverse is the difference between the sum of
the measured angles and the geometrically correct total for the polygon. Checks will be
considered for both polygon and link theodolite traverses.
Check for polygon traverse:
The observed angles of a polygon traverse can be either internal or external angles.
For internal angles: Sum of internal angles = (n – 2) x 180º
For external angles: Sum of external angles = (n + 2) x 180º
Where n is the number of angles measured.
Check for link traverse:
Sum of angles = (final forward bearing – initial back bearing) + (n – 1) x 180º
where n is the number of angles measured between the initial back bearing and
final forward bearing.
Considering the above example, we will have the sum of angles as shown in Table 4.1.
Table 4.1: Sum of included angles
Station
Observed angle
A
115º 11' 20''
B
95º 00' 20''
C
129º 49' 20''
D
130º 36' 20''
E
110º 30' 00''
F
138º 54' 40''
Sum
720º 02' 00''
48
Required sum of internal angles = (n – 2) x 180º = (6 – 2) x 180º = 720º 00' 00''
Hence, Angular misclosure = 720º 02' 00'' - 720º 00' 00'' = + 02' 00''
The allowable misclosure (E) is E   KS N , where N is the number of traverse stations, S is
the smallest reading interval on the theodolite in seconds, and K is a multiplication factor of 1
to 3, depending on weather conditions, number of rounds taken, etc.
Step 2: Adjustment of angles
The angles are adjusted by distributing the misclosure over all the measured angles as in
Table 4.2. For the above, the adjustment per angle will be – (02' 00'')/6, since six (6)
internal angles were measured:
Table 4.2: Adjustment of angles
Station
Observed angle
Adjustment
Adjusted Left-hand angle
A
115º 11' 20''
-20''
115º 11' 00''
B
95º 00' 20''
-20''
95º 00' 00''
C
129º 49' 20''
-20''
129º 49' 00''
D
130º 36' 20''
-20''
130º 36' 00''
E
110º 30' 00''
-20''
110º 29' 40''
F
138º 54' 40''
-20''
138º 54' 20''
Sum
720º 02' 00''
- 02' 00''
720º 00' 00''
Step 3: Conversion of adjusted angles to whole circle bearings
The conversion of the adjusted angles to bearings for the traverse is illustrated in Figure
4.3. This is done because the angles measured by the theodolite are not bearings. The
whole circle bearing of a line is the clockwise horizontal angle measured from the meridian
passing through the starting point of the line to the line.
N
b
aX
49
Figure 4.3: Whole circle bearing calculation (Source: Uren and Price 1994)
From Figure 4.3:
Forward bearing = back bearing of previous leg+ left-hand angle
Considering the above example:
50
N
F
7
N
A
1
D
i
r
e
0
B
9
C
F
i
51
At station A:
Forward bearing AB = back bearing of FA + left-hand angle at A
= forward bearing of AF + left-hand angle at A
= 70º 00' 00'' (given) + 115º 11' 00''
= 185º 11' 00''
At station B:
Forward bearing BC = back bearing of AB + left-hand angle at B
= forward bearing of BA + left-hand angle at B
= 05º11'00'' + 95º 00' 00''
= 100º 11' 00''
The bearings of all the other lines are computed in the same manner, and are shown in Table
4.3.
Table 4.3: Computation of Adjusted Forward Bearings
Stations
Back Bearing of
Adjusted Left-
Adjusted
previous leg
hand angle
Forward Bearing
From
To
F
A
A
B
70° 00′ 00″
B
C
05° 11′ 00″
Traverse Leg
250° 00′ 00″
FA
115° 11′ 00″
185° 11′ 00″
AB
95° 00′ 00″
100° 11′ 00″
BC
52
C
D
280° 11′ 00″
129° 49′ 00″
50° 00′ 00″
CD
D
E
230° 00′ 00″
130° 36′ 00″
00° 36′ 00″
DE
E
F
180° 36′ 00″
110° 29′ 40″
291° 05′ 40″
EF
F
A
111° 05′ 40″
138° 54′ 20″
250° 00′ 00″
FA
Note: A left-hand angle at a traverse station (also termed angle-to-the-right) is the horizontal
angle measured clockwise from a backsight on the “rearward” traverse station to a
foresight on the “forward” traverse station.
For polygon traverses when working in an anticlockwise direction around the traverse, the
left-hand angles are the internal angles of the traverse and when working in a clockwise
direction, the left-hand angles will be the external angles.
Step 4: Calculating Latitudes and Departures (L&D) of traverse Legs
The latitude of a traverse leg or a line is the distance the line extends in a north or south
direction, and is equal to the length of the line multiplied by the cosine of its whole circle
bearing. A line running in a northerly direction has a plus latitude; one running in a southerly
direction has a minus latitude. Latitude is also termed change in northing or southing of the
end points of a line.
The departure of a traverse leg or a line is the distance the line extends in an east or west
direction, and is equal to the length of the line multiplied by the sine of its whole circle bearing.
A departure to the east is considered plus; a departure to the west is minus. Departures are
sometimes termed difference in easting or westing of the end points of the line.
53
N
∆
E
∆
N
B
L
A
0
E
D
e
Figure 4.5: Computation of Latitudes and Departures
From the rectangular grid system shown in Fig. 4.5,
i.
Latitude of line AB = change in northing of line AB, N  Lcos
ii.
Departure of line AB = change in easting of line AB, E  Lsin
where L is the horizontal length of the traverse leg and
 is the whole circle bearing of the
traverse leg. In traverse computations, north latitudes and east departures are considered
plus whilst south latitudes and west departures are considered minus.
Step 5: Latitude and Departure Closure Conditions
If all angles and distances are measured perfectly,
54
(i)
For a closed-polygon traverse, the algebraic sum of latitudes of all the legs should
equal zero since the traverse starts and finishes on the same point. Likewise, the
algebraic sum of all departures should equal zero. That is,
 E  0 and  N  0
(ii)
For a closed-link traverse, the algebraic sum of latitudes should equal the total
difference in latitude between the starting and ending control points. The same
condition applies to departures in a link traverse. That is,
 E  E
Y
 EX
and
 N  N
Y
 NX
where stations Y and X are the final and starting points respectively of the traverse. Since
Y and X are control points, their coordinates are known, and therefore the values of
EY  EX and NY  N X can be calculated.
Step 6: Traverse Linear Misclosure and Fractional Linear misclosure
Because the measurements are not perfect, and errors exist in the angles and distances, the
above conditions rarely occur. The amounts by which they fail to be met are termed latitude
misclosure, eN and departure misclosure, eE . Their magnitudes or values are computed by
algebraically summing the latitudes and departures, and comparing the totals to the required
conditions.
The linear misclosure e is given by
e  (latitude misclosure) 2  (departure misclosure) 2
 (eN2  eE2 )
Fractional Misclose (F.M) of the traverse is given by
F.M .  1 in (total length of traverse legs linear msclosure)
55
Table 4.4: Computation of Latitudes and Departures and Rectangular Coordinates of Traverse Stations
Latitudes (m)
Horizontal
Departures (m)
Cor.
Distance
Coordinates (m)
Cor.
From
W.C.B.
(m)
N
Cor
N
E
Cor
E
N
E
To
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
1000.00
500.00
A
A
185° 11′ 00″
429.37
-427.61
0.02
-427.59
-38.79
-0.04
-38.83
572.41
461.17
B
B
100° 11′ 00″
656.54
-116.08
0.03
-116.05
646.20
-0.05
646.15
456.36
1107.32
C
C
50° 00′ 00″
301.83
194.01
0.01
194.02
231.22
-0.03
231.19
650.38
1338.51
D
D
00° 36′ 00″
287.40
287.38
0.01
287.39
3.01
-0.02
2.99
937.77
1341.50
E
E
291° 05′ 40″
526.72
189.57
0.03
189.60
-491.42
-0.04
-491.46
1127.37
850.04
F
F
250° 00′ 00″
372.47
-127.39
0.02
-127.37
-350.01
-0.03
-350.04
1000.00
500.00
A
Σ
2574
-0.12
+0.12
0
+0.21
-0.21
0
Step 7: Adjustment of the Latitudes and Departures
For any closed traverse, the linear misclosure must be adjusted (or distributed) throughout the
traverse to close or balance the figure. The Compass or Bowditch Rule, which adjusts the
latitudes and departures of the traverse legs in proportion to their lengths, is normally used.
Correction in latitude for leg AB =
-(total latitude misclosure)
 length of AB
total traverse length
Correction in departure for leg AB =
-(total departure misclosure)
 length of AB
total traverse length
56
Step 8: Computation of rectangular coordinates
Northing N B and Easting EB coordinates of a forward station B are given by,
N B  N A  latitude of AB
EB  E A  departure of AB
where station A is the instrument (or starting) station.
Questions
1. The internal angles of a closed-loop anticlockwise traverse ABCDEA were observed with a
10'' theodolite as follows: A = 89º 39' 40'', B = 129º 18' 50'', C = 83º 41' 00'', D = 112º 38' 10'',
and E = 124º 43' 10''. The bearing of line AE is fixed at 254º 58' 00'' and the coordinates of A
are (1000.00 m E, 500.00 m N).
The following horizontal distances were measured: AB = 120.39, BC = 232.71, CD = 168.30,
DE = 152.00 and EA = 186.13 m. Calculate:
(i)
the angular misclosure of the traverse and adjust the angles
(ii)
the forward bearings of the traverse legs
(iii)
the coordinate differences for the traverse adjusting any linear misclosure by the
Bowditch method
(iv)
the fractional linear misclosure (FLM)
the coordinates of stations B, C, D, and E.2. The measured angles and lengths of a clockwise
traverse CDEF are given in Table 4.5. Assuming the coordinates of station C to be (250.00 m E,
250.00 m N) and that station D is due East of C, calculate the adjusted coordinates of stations
D, E and F, and the fractional linear misclose of the traverse.
57
Table 4.5
Angle
Observed external angle
Leg
Horizontal length (m)
FCD
272º 55'
FC
112.61
CDE
296º 43'
FE
106.26
DEF
262º 13'
CD
180.29
EEC
248º 13'
DE
164.58
3. A four-sided closed-loop traverse WARDW, lettered anticlockwise, si set out on a
construction site and the angles and lengths shown in Table 4.6 were measured. The
coordinates of station W are (500.00 m E, 500.00 m N) and the whole circle bearing of line WA
is 180º 00' 00''.
Calculate the coordinates of stations A, R, and D, adjusting any misclosure by the Bowditch
method.
Table 4.6
Internal angle
Horizontal length (m)
W
110º 47' 10''
WA = 65.39
A
91º 25' 30''
AR = 285.94
R
111º 20' 30''
RD = 233.57
D
46º 26' 10''
DW = 402.45
58
5 Computation of Areas
5.1 Reasons for computing areas of tracts of land
(i) To state the acreage of a parcel of land in the deed describing the property
(ii) To determine the acreages of fields, lakes or the number of square metres to be surfaced,
paved, or seeded
(iii) To determine end areas for earthwork volume calculation
In plane surveying, area is considered to be the orthogonal projection of the surface onto a
horizontal plane. The commonest unit of area for large tracts is the acre. 1 acre = 43,560
square feet = 4046.835 square metres. 1 hectare = 10,000 square metres = 2.471 acres.
5.2 Methods of area computation
The area of any surveyed figure can be calculated from either (1) the field measurements or
(2) the plotted plan. The surveyed figure or plotted plan may be straight-sided, irregular-sided
or a combination of both.
Field measurements methods are the more accurate and include:
(i) division of the tract into simple geometric figures such as triangles, rectangles, and
trapezoids;
(ii) offsets from a straight line;
(iii) cross-coordinates; and
(iv) double-meridian distances.
Methods of determining area from plotted plans include:
59
(i) counting coordinate squares;
(ii) dividing the area into triangles, rectangles, or other regular geometric shapes;
(iii) digitizing coordinates;
(iv) running a planimeter over the enclosing lines
Since plans themselves are derived from field measurements, methods of area determination
invariably depend on this basic source of data.
5.2.1 (i) Areas by subdivision into simple figures
The tract is divided into simple geometric figures such as triangles, rectangles, or trapezoids.
The sides and angles of these figures are then measured in the field and their individual areas
calculated and totalled.(1) Areas from triangles
(a) Semi-perimeter method
The area of a triangle whose sides are known is computed by the formula
area  s ( s  a )( s  b)( s  c)
s
1
(a  b  c)
2
where s is the semi-perimeter and a , b and c are the lengths of the sides of the
triangle.
(b) Area of a triangle with known height
area 
1
(base x perpendicular height )
2
(c) Area of a triangle with known included angle
area 
1
ab sin C
2
where C is the angle included between sides a and b
60
Worked example 1:
Compute the area of a right-angled triangle of sides 3 m, 4 m and 5 m.
S
o
c
b
a
F
i
1. Using the first formula,
area  s( s  a)( s  b)( s  c) ; s 
1
1
(a  b  c)  (4  3  5)  6m
2
2
area  6(6  4)(6  3)(6  5)  36m 4  6m 2
2. Using the second formula,
61
area 
1
1
(base x height )  (4 m x 3 m)  6 m 2
2
2
3. Using the third formula,
area 
1
1
ab sin C  x 4m x 3m x sin 90  6 m 2
2
2
area 
1
(a  b) h
2
(2) Area from a trapezium
where a and b are parallel sides and h is the perpendicular distance or height between the
parallel sides.
Worked example 2: Compute the area of a trapezium whose parallel sides are 4m and 6m and
of height 5m.
62
a
b
a
h
h
b
63
Solution to example 2:
area 
1
1
(a  b)h  (4m  6m)5m  25 m 2
2
2
5.2.1 (ii) Areas by offsets from straight lines
Irregular tracts can be divided into a series of trapezoids by measuring right-angle offsets from
points along a measured reference line. The spacing between the offsets may either be regular
or irregular, depending on conditions. For regularly spaced offsets, trapezoidal and/or
Simpson’s rules are applied.
5.2.1 (ii a) Trapezoidal rule
The trapezoidal rule approximates an irregular boundary of a tract to a series of straight lines.
A reference traverse line close to the boundary is selected and offsets (or ordinates)
y1 , y2 , y3 ,... yn at regular intervals (or width) w are measured from the traverse line to the
boundary.
64
I
r
G
O
f
A
T
y
Y
Y
1
2
3
w
w
w
Y
y
Y
Y
5
-
n
n
w
w
w
B
T
The area bounded by the traverse line, the irregular boundary and the first and last offsets is
computed using the trapezoidal rule given as follows:
65
Area  Regular Interval x  Average of first and last offsets  sum of other offsets 
1

 w   y1  yn   y2  y3  ... yn 1 
2

w
  y1  yn   2  y2  y3  ... yn 1  
2
Note:
i.
If the first or last offset is zero, it must still be included in the computation.
ii.
The number of offsets could be odd or even.
iii.
The offsets must be at regular intervals.
iv.
The interval must be short enough for the length of boundary between the offsets
to be assumed straight.
v.
If the boundary is curved to such an extent that approximating it with a series of
straight lines would introduce appreciable error, Simpson’s rule should be used.
5.2.1 (ii b) Simpson’s rule
Simpson’s rule approximates the boundary to a series of parabolic arcs. A reference traverse
line close to the boundary is selected and odd number of offsets (or ordinates) y1 , y2 , y3 ,... yn
at regular intervals (or width) w are measured from the traverse line AB to the boundary GT.
I
r
G
O
f
T
y
1
A
y
2
y
3
y
4
y
y
y
-
n
n
y
n
B
N
o
t
66
Simpson’s rule states that the area enclosed by a curvilinear figure divided into an even
number of strips of equal width is equal to one-third the width of a strip, multiplied by the sum
of the two extreme offsets, twice the sum of the remaining odd offsets, and four times the sum
of the even offsets.
Area 
Regular Interval ( sum of first and last offsets)  4( sum of even offsets) 

 2( sum of remaining odd offsets) 
3

Note: Simpson’s rule can be applied to an odd number of offsets only.
Worked example 3:
B
o
O
f
R
0
1
0
2
0
3
0
4
0
5
0
6
0
7
0
P
Calculate the area between the traverse line RP and the irregular boundary line. The offset
distances from the station R to P are 3.0, 8.0, 10.0, 9.5, 9.2, 7.1, 4.5 and 2.5 m respectively.
Solution to example 3
67
Since there is an even number of offsets between R and P at regular intervals of 10 m,
Trapezoidal rule could be used.
interval
 first offset  last offset  twice sum of all other offsets 
2
10
 3.0  2.5  2  8.0  10.0  9.5  9.2  7.1  4.5  
2
 510.5 m 2
Area 
Worked example 4
The following offsets 10 m apart were measured at right angles from a traverse line PQ to an
irregular boundary: 4.0 m, 4.5 m, 5.1 m, 6.5 m, 6.3 m, 5.1 m and 4.0 m respectively. Calculate
the area between the line PQ and the irregular boundary line.
Solution to example 4
Using Simpson’s rule,
Area 
Regular Interval ( sum of first and last offsets )  4( sum of even offsets) 

 2( sum of remaining odd offsets) 
3

10
 4.0  4.0   4  4.5  6.5  5.1  2  5.1  6.3 
3 
 317.33 m2

Worked example 5
State Simpson’s rule for the determination of areas.
In a chain survey the following offsets were taken to a fence from a chain line:
Chainage (m)
0
20
40
60
80
100
120
140
160
180
Offset (m)
0
5.49
9.14
8.53
10.67
12.50
9.76
4.57
1.83
0
Find the area between the fence and the chain line using
(a) Trapezoidal rule
68
(b) Simpson’s rule
Solution to example 5
(a)
Using Trapezoidal rule,
interval
 first offset  last offset  twice sum of all other offsets 
2
20

0  0  2  5.49  9.14  8.53  10.67  12.50  9.76  4.57  1.83 
2 
 1249.80 m 2
Area 
(b) Since Simpson’s rule can be applied to an odd number of offsets only, it will be used here
to calculate the area contained between the first and ninth offsets. The residual triangular
area between the ninth and tenth offsets is calculated separately.
Area 
Regular Interval ( sum of first and last offsets)  4( sum of even offsets) 

 2( sum of remaining odd offsets) 
3

It is often convenient to tabulate the working.
Offset No.
Offset
Simpson’s Multiplier
Product
1
0
1
0
2
5.49
4
21.96
3
9.14
2
18.28
4
8.53
4
34.12
5
10.67
2
21.34
6
12.50
4
50.00
69
7
9.75
2
19.50
8
4.57
4
18.28
9
1.83
1
1.83
Sum=185.31
The area between offset 1 and 9 is therefore given by
Area(19) 
20
185.31  1235.40 m2
3
Now the area between the ninth and tenth offsets is computed
Area(910) 
20
x 1.83  18.30 m 2
2
Therefore, the total area between the chain line and the fence is 1253.70 m 2 .
5.22Area by coordinates
Computation of areas within closed polygons is most frequently done using the cross
coordinate method. In this procedure, coordinates of each angle point in the figure must be
known, and the following steps taken:
i.
list the X and Y (or Northing and Easting) coordinates of each point in succession in
two columns, with coordinates of the starting point repeated at the end;
ii.
Multiply the northing (X) of each station by the easting (Y) of the succeeding station
and find the sum. Consider the sum plus;
iii.
Multiply the easting (Y) of each station by the northing (X) of the succeeding station
and find the sum. Consider the sum minus;
iv.
Find the algebraic summation of (ii) and (iii) above, and divide its absolute value by
2 to get the area.
70
For four traverse stations numbered 1 up to 4, the area enclosed by the polygon is given by
Area 
1
 N1 E2  N 2 E3  N 3 E4  N 4 E1   ( E1 N 2  E2 N 3  E3 N 4  E4 N1 ) 
2
Worked example 6: Given that the coordinates of the traverse stations A, B, C and D are as
follows;
Travesre Station
Northing, X (m)
Easting, Y (m)
A
100.0
200.0
B
205.0
300.0
C
250.0
350.0
D
200.0
400.0
Calculate the area in hectares enclosed by the stations
Solution to example 6:
Cross coordinate
Traverse Northing,
Easting,
Station
X (m)
Y (m)
A
100.0
200.0
B
205.0
300.0
100.0 x 300.0 = 30,000.0
200.0 x 205.0 = 41,000.0
C
250.0
350.0
205.0 x 350.0 = 71,750.0
300.0 x 250.0 = 75,000.0
D
200.0
400.0
250.0 x 400.0 = 100,000.0
350.0 x 200.0 = 70,000.0
A
100.0
200.0
200.0 x 200.0 = 40,000.0
400.0 x 100.0 = 40,000.0
Plus ( N n En 1 )
Minus ( En N n 1 )
71
Sum = 241,750.0
Sum =226,000.0
Algebraic sum = 241,750.0 – 226,000.0 = 15,750.0
algebraic sum 15, 750.0

 7,875 m2
2
2
2
7,875 m

x hectare  0.7875 ha
10, 000 m2
Area 
5.3 Area by measurements from maps or plans
To determine the area of a tract of land from map measurements, its boundaries must first be
identified on an existing map or a plot of the parcel drawn from survey data. Then any one of
the methods described under the following subsections can be used to determine its area.
5.3.1 Area by counting coordinate squares
This method involves overlaying the mapped parcel with a transparency having a superimposed
grid and counting the number of grid squares included within the tract. Partial squares are
estimated and added to the total. The area is obtained by multiplying the total number of
squares counted by the area represented by each square.
5.3.2 Area by scaled lengths
The boundary of the parcel is first identified on the map. The tract is then divided into
triangles, rectangles, or other regular figures and the sides are measured. The area of each
regular figure is then computed using standard formulas and totalled.
5.3.3 Area by digitizing coordinates
The map or plan containing the parcel whose area is required is placed on a digitizing table
which is interfaced with a computer, and the coordinates of its corner points quickly and
72
conveniently recorded. From the file of coordinates, the area is computed using the cross
coordinate method.
5.3.4 Area by planimeter
A planimeter measures the area contained within any closed figure that is circumscribed by its
tracer. Measurement of areas by planimeter is also referred to as measurement of areas by
mechanical integration. The planimeter is thus a mechanical integrator used for the
measurement of the areas (of all shapes) from plans. It consists essentially of
a) The pole block (which is fixed in position on the plan by a fine retaining needle).
b) The pole arm (which is pivoted about the pole block at one end, and carries the
integrating unit at the other end).
c) The tracing arm, attached at one end to the integrating unit, and carrying at the other
end the tracing point or optical tracer.
d) The measuring unit, consisting of a hardened steel integrating disc carried on pivots. A
primary drum, divided into 100 parts, is directly connected to the disc spindle, and
readings up to 1/1000th of a revolution of the integrating disc are obtained, either by
estimation with reference to an index mark, or by vernier on an opposite drum. Another
indicator gives the number of complete revolutions of the disc.
A planimeter is most useful for irregular areas. It is simple to use, and capable of a high degree
of accuracy. The planimeter, shown in Figure 3-12, touches the paper at three points:
the anchor point, P;
the tracing point, T; and
the roller, R.
73
The adjustable arm, A, is graduated to permit adjustment to the scale of the plot. This
adjustment provides a direct ratio between the area traced by the tracing point and the
revolutions of the roller. As the tracing point is moved over the paper, the drum, D, and the
disk, F, revolve. The disk records the revolutions of the roller in units of tenths; the drum, in
hundredths; and the vernier, V, in thousandths.
To use it, place the pole block in a suitable position relative to the figure such that the tracing
point can reach every part of the outline. Put the tracing point on a known point on the outline
and read both drums with reference to the index mark.
Carefully move the tracing point in a clockwise direction round the outline of the figure, back to
the starting point. Take a second reading. The different between the two readings, multiplied
by the scale factor, gives the required area. Repeat the operation until three consistent values
are obtained. The mean of the three is then taken as the accepted value.
[Note: the foregoing refers to conventional mechanical planimeters. There are now, on the
market, Digital Planimeters, shown in which are more versatile, yet simple to use.
An
electronic planimeter operates similarly to the mechanical type, except that the results are
given in digital form on a display console].
74
Fig. 8.1 Digital Planimeter
5.4 Sources of error in determining areas
Some sources of error in area computations are:
i.
Errors in the field data from which coordinates and/or plans are derived.
ii.
Making a poor selection of intervals and offsets to fit a given irregular boundary
properly.
iii.
Making errors in scaling from plans/maps.
iv.
Shrinkage and expansion of plans/maps.
v.
Using coordinate squares that are too large and therefore make estimation of areas
of partial blocks difficult.
vi.
Making an incorrect setting of the planimeter scale bar.
vii.
Forgetting to divide by 2 in the cross coordinate method.
75
6 Levelling and Contouring
Levelling is the process of determining the relative elevations (or height differences) of points
on the earth’s surface. A level instrument, a barometer or a theodolite may be used to
determine the relative elevations of points.
Levelling results are used to:
(i)
design highways, railways, canals, sewers, water supply systems, and other
facilities having grade lines that best conform to existing topography;
(ii)
lay out construction projects according to planned elevations;
(iii)
calculate volumes of earthworks and other materials;
(iv)
investigate drainage characteristics of an area;
(v)
develop maps showing general ground configurations; and
(vi)
study earth subsidence and crustal motion.
6.1 Basic definitions
Horizontal line at
point A
A
Elevation of point A
(measured along
vertical line through
A)
Level line
MSL datum
Figure 6. 1: Levelling terminolgies
76
(i)
Vertical line
A vertical line is a line that follows the direction of gravity as indicated by a plumb line. A
plumb line is the direction in which gravity acts.
(ii)
Horizontal plane or line
A horizontal plane is a plane perpendicular to the direction of gravity. A horizontal line is a
line in a horizontal plane or a straight line perpendicular to a vertical line.
(iii)
Level line: A level line is a curved line in a level surface.
(iv)
Level surface
A level surface is a curved surface that is perpendicular to the local plumb line at every
point. Level surfaces are approximately spheroidal in shape.
(v)
Vertical datum: Any level surface to which elevations are referred (e.g. mean sea
level).
(vi)
Elevation
An elevation is a vertical distance above or below a reference vertical datum. In surveying,
the reference datum that is universally employed in levelling is that of the mean sea level
(MSL)
(vii)
Mean Sea Level (MSL)
MSL is the average height of the sea’s surface for all stages of the tide over a considerable
period.
(viii) Bench Mark (BM)
A bench mark is a relatively permanent object, natural or artificial, having a marked point
whose elevation above or below an adopted datum is known or assumed.
77
(ix)
Vertical control
A series of bench marks or other points of known elevations established throughout
an area.6.2 Curvature and refraction
It is essential to understand the nature of the earth’s curvature and atmospheric refraction as
they affect levelling operations.
The definition of a level surface indicates that it is parallel to the curvature of the earth. A line
of constant elevation, termed a level line, is likewise a curved line and is everywhere normal to
the plumb line. However, a horizontal line of sight through the surveyor’s telescope is
perpendicular to the plumb line only at the point of observation.

Curvature
A
h
o
l
e
m
e
v
e
(a)
d
C
R
R
78
(b)
Figure 6.2: Earth's curvature, (a) and (b)
Figure 6.2 shows a section passing through the earth’s Centre. It can be seen that the level
and horizontal lines through the instrument diverge. This divergence between the level line and
horizontal line over a specified distance is known as the curvature, C. This is caused by level
lines following the curvature of the earth which is defined by the mean sea level.

Refraction
When considering the divergence between level and horizontal line, one must also account for
the fact that all sight lines are refracted downward by the earth’s atmospheric conditions. That
is, the effect of atmospheric refraction on a line of sight is to bend it towards the earth’s
surface causing staff readings to be too low.
Note: The combined effects of curvature and refraction are negligible when undertaking
levelling where the sighting length is less than 120 metres. However, if longer sight length
must be used, the effects of curvature and refraction will cancel out if the sight lengths are
equal.
6.3 Fieldwork in levelling
Levelling between two points
This is the basis for all levelling work no matter how complex the particular levelling may
be.
l
i
0
2
I
A
9
1
B
I
2
C
79
Figure 6. 3: Principles of levelling
Referring to figure 6.9, the direction of levelling is from A to B and to C:
A is known as the back sight (BS) and B is known as the fore sight (FS)
if level
instrument is at I1.
B is known as the back sight and C is known as the fore sight if level instrument is at I2.
On and on it goes.
Point B is called a change point (CP) since at point B a foresight is taken followed by a
backsight reading.
In practice, a BS is the first reading taken after the instrument has been set up and is
always to a point of known or calculated reduced level. Conversely, a FS is the last reading
taken before the instrument is moved. Any readings taken between the BS and FS from the
same instrument position are known as intermediate sights (IS).
2.500m and 0.500m are the respective readings observed on the staff at stations A and B.
In all cases, the back-station is observed on a known reduced level (RL).
Back-sight (BS) = 2.500m
Foresight (FS)
= 0.500m
If RL at A
= RLA = 95.400m
Then RL at B
= RLB = 95.400 + (2.500 - 0.500) = 97.400m
In general, the reduced level is given by:
RLB = RLA + (BS – FS)
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6.4 Field booking and reduction of levels
Figure 6. 4: Levelling and booking sequence
Figure 6.4 will be used in demonstrating the field bookings and the reduction of the levels. Two
methods of booking and reduction of levels will be discussed, i.e. the rise and fall method and
the height of collimation method.
(i) Rise and fall method
Table 6. 1: Rise and fall method
BS
IS
FS
Rise
Fall
Initial RL
Adjustment
Adjusted RL
Remarks
TBM
2.191
49.873
0
49.873
49.873
49.559
+0.002
49.561
A
0.180
49.739
+0.002
49.741
B
0.829
50.568
+0.002
50.570
C (CP)
0.506
51.074
+0.004
51.078
D
50.776
+0.004
50.780
E (CP)
2.505
0.314
2.325
3.019
1.496
2.513
1.752
2.811
0.298
81
TBM
3.824
6.962
8.131
8.131
1.515
2.072
48.704
2.684
48.704
2.684
49.873
-1.169
-1.169
+0.006
48.710
48.710
1.169
Checks:
Checks to be performed for the rise and fall method are as follows:
∑ (BS) - ∑ (FS) = ∑(Rises) -∑(Falls)
= Last RL – First RL
Adjustment:
Adjustments to be performed for the rise and fall method are as follows:
The difference between the calculated and known values of the RL of the final Bench Mark
(BM) is -0.006m. This is known as the misclosure.
Since the misclosure is -0.006m, it implies that the total adjustment is +0.006m. This must be
distributed to all the RL’s.
Correction per station = -(misclose divided by number of instrument stations or backsights).
Since there are three (3) stations, +0.002m (+0.006/3) is added to the reduced levels found
from each instrument position. All reduced levels derived from the same instrument station
receive the same value of correction.
Correction Cn to be applied to a reduced level obtained from the nth instrument station is given
by
Cn = n x correction per station
These cumulative adjustments are done in Table 6.1.
(ii) Height of Collimation method.
The basic idea here is to determine the height of plane of collimation (HPC) for all the
instrument stations (change points). This is then used in calculating the RL’s of the various
stations.
82
Table 6.2: Height of Collimation method
Adjustmen
BS
IS
FS
HPC
Initial RL
t
Adjusted RL
Remarks
52.064
49.873
0
49.873
TBM 49.873
2.505
49.559
+0.002
49.561
A
2.325
49.739
+0.002
49.741
B
50.568
+0.002
50.570
C (CP)
51.074
+0.004
51.078
D
50.776
+0.004
50.780
E (CP)
48.704
+0.006
48.710
TBM 48.710
2.191
3.019
1.496
53.587
2.513
1.752
2.811
52.528
3.824
106.11
6.962
7.343
8.131
5
300.420
8.131
7.343 + 8.131 + 300.420 = 315.894
-1.169
(52.064 x 3) + (53.587 x 2) + 52.528 = 315.894
6.962
48.704
8.131
49.873
-1.169
-1.169
Calculation for HPC’s and RL’s:
Calculations to be performed for the height of collimation method are as follows:
For instrument station I1, HPC = 49.873 + 2.191 = 52.064m
RL of A = 52.064 – 2.505 = 49.559m
RL of B = 52.064 – 2.325 = 49.739m
83
RL of A = 52.064 – 1.496 = 50.568m
For instrument station I2, HPC = 50.568 + 3.019 = 53.587m
The same procedure is continued for the other stations.
Checks:
Checks to be performed for the height of collimation method are as follows:
∑ (BS) - ∑ (FS) = Last RL – First RL
∑ (IS) + ∑ (FS) + ∑(RL’s except first) = ∑(each HPC x number of applications)
The critical difference between rise and fall and height of collimation methods:

The rise and fall method is quicker to reduce where a lot of backsights and foresights have
been taken and very few intermediate sights taken. For this reason, the rise and fall
method tends to be used when establishing control when no intermediate sights would
normally be taken.

The collimation method is quicker to reduce where a lot of intermediate sights have been
taken since fewer calculations are required and it is a good method to use when setting
out levels where, usually, many readings are taken from each instrument position. A
disadvantage of this method is that the check can be lengthy.
6.5 Precision of levelling
For levelling, the allowable misclosure is given by;
Allowable misclosure = ± 5√n mm, where n is the number of instrument positions.
When the actual and allowable misclosures are compared and it is found that the actual value
is greater than the allowable value, the levelling should be repeated. If, however, the actual
value is less than the allowable value, the misclosure should be distributed equally between the
instruments positions as already described.
84
6.6 Inverted staff
Frequently the reduced levels of points above the height of instrument are required. For
example: the soffit of a bridge or underpass; the underside of a canopy; the levels of roof,
caves, buildings etc. Generally, these points will be above the line of collimation. To obtain the
reduced levels of such points, the staff is held upside down in an inverted position with its base
on the elevated point.
When booking an inverted staff reading it is entered in the levelling table with a minus sign.
The calculation proceeds in the normal way, taking the minus sign into account.
6.7 Contouring
Contouring is art of showing relief features on a flat sheet of paper. This is because both the
height and shape of the land surface can be specifically shown.
Contours are lines drawn on maps to join places of equal heights above or below sea level.
For example, on a topographic map, a 500-ft contour line means the line joins points on the
land that are 500 ft. above sea level.
6.7.1 Characteristics of contour lines
(i)
Contour lines are continuous lines within the areas they cover;
(ii)
Contour lines do not meet or cross each other;
(ii)
A contour line do not split into other contours or join any other contour line;
(iii)
Widely spaced contours indicate a gentle slope;
(iv)
Contours which are closer together indicate steep slopes and contours which are
evenly spaced indicate uniform slope.
Characteristically, contours are indicated by brown curved lines on maps. Every fifth contour
line is thickened to facilitate easy reading. The interval between two contours, i.e. vertical
interval (VI), is constant on the same map. Contour maps are used in obtaining sections (i.e.
cross sections and longitudinal sections). Sectioning is usually undertaken for construction work
85
such as road works, railways and pipelines. Two types of section are often necessary and these
are longitudinal and cross sections. A longitudinal section (or profile) is taken along the
complete length of the proposed centre line of the construction showing the existing ground
level. Cross sections are taken at right angles to the centre line such that information is
obtained over the full width of the proposed construction.
6.7.2 Contour Interval
The difference in height between successive contours is known as the contour or vertical
interval and this interval dictates the accuracy to which the ground is represented.
6.7.3 Gradients
The gradient of the ground between two points is given by:
Gradient =
Vertical Interval
Horizontal Equivalent
For example, using figure 6.12, the gradient of the ground between the points A and C is:
= AB =
BC
D
E
Vertical Interval
Horizontal Equivalent
A
C
4
3
2
1
0
6
4
86
2
A
B
C
Figure 6. 5: Gradient and profile
6.7.4 Plotting Contours
Contours can be plotted directly or indirectly.

Direct contouring:
In this method, the position of particular contour is located on the ground and marked
using a level instrument. Using this method, the contour lines are physically followed on the
ground. Two distinct operations can be used in this method:
a. Levelling
The height of collimation (= reduced level at observed station + backsight reading at
observed station) of the levelling instrument is first of all determined. Staff-man holds
the staff facing the instrument and backs slowly until prompted by observer for a
particular reading on the staff. Pegs are positioned are the various prompted points
which will indicate a particular contour line.
b. Survey of the pegs
Here the plan positions of the pegs are established to allow plotting to take place. On a
smaller site, chain lines and offset can be used to survey these pegs. On a larger site,
compass, Theodolite or tachometric traverse can be employed. The plan positions of the
contours are then plotted directly onto the site plan and smooth curves drawn through
them.

Indirect contouring:
87
When using this method, no attempt is made to follow the contour lines. Instead, a series
of spot levels is taken and the contour positions are interpolated. Three distinct positions
are involves:
a. Setting out grid
In this method, the area to be contoured is carefully divided into a series of lines that
form squares. The interval of the grid lines depends on the contour interval.
The area to be contoured is divided into a series of lines forming squares and ground
levels are taken at the intersection of the grid lines. The sides of the squares can vary
from 5 to 30 m, the actual figure depending on the accuracy required and on the nature
of the ground surface. The more irregular the ground surface the greater the
concentration of grid points.
b. Levelling
Levels are taken at the intersections of the grid lines. The reduced level of every point
on the grid is obtained.
To obtain the ground level at each grid point the person holding the staff lines the staff
in with the two ranging rods in each direction that intersect at the point being levelled,
and a reading is taken. The procedure is repeated at all grid points.
c. Interpolating the contours
This is done mathematically or graphically. With the mathematical method, the positions
of the contours are interpolated mathematically from the reduced level by simple
proportions. The height difference between each spot height is calculated and used with
the horizontal distance between them to calculate the position on the line joining the
spot heights at which the required contour is located
6.2.7 Use of contour maps
Contour maps are used in obtaining sections (i.e. cross sections and longitudinal sections). For
example, it is possible to use contours to obtain sectional information for use in the initial
88
planning of such projects as roads, pipelines, earthworks and reservoirs. These sections are
used in earthworks.
6.3 Sectioning
Contour maps are used in obtaining sections (i.e. cross sections and longitudinal sections).
Sectioning is usually undertaken for construction work such as road works, railways and
pipelines. Two types of section are often necessary and these are longitudinal and cross
sections. A longitudinal section (or profile) is taken along the complete length of the proposed
Centre line of the construction showing the existing ground level. Cross sections are taken at
right angles to the Centre line such that information is obtained over the full width of the
proposed construction.
6.3.1 Longitudinal sections
In surveying, a longitudinal section (or profile) is taken along the complete length of the
proposed centre line of the construction showing the existing ground level. Cross sections
taken at right angles to the Centre line such that information is obtained over the full width of
the proposed construction. Levelling can be used to measure heights at points on the centre
line so that the profile can be plotted.
Generally, this type of section provides data for determining the most economic formation
level, this being the level to which existing ground is formed by construction methods. The
fieldwork in longitudinal sectioning normally involves two operations.

Firstly, the centre line of the section must be set out on the ground and marked with
pegs. For most works, this is done by theodolite and some form of distance
measurement so that pegs are placed at regular intervals (frequently 20 m) along the
centre line.

Secondly as soon as the centre line has been established levelling can commence.
For longitudinal sections, it is usually sufficiently accurate to record readings to the nearest
0.01 m. Levels are taken at the following points, the object being to survey the ground profile
as accurately as possible:
89
1. At the top and ground level of each centre line peg noting the through chainage of the
peg.
2. At points on the centre line at which the ground slope changes.
3. Where features cross the centre line, such as fences, hedges, roads, pavements, ditches
and so on. At points where, for example, roads or pavements cross the centre line,
levels should be taken at the top and bottom of kerbs. At ditches and streams, the
levels at the top and bottom of any banks as well as bed levels are required.
4. Where necessary, inverted staff readings to underpasses and bridge soffits would be
taken.
In order to be able to plot levels obtained in addition to those taken at the centre line pegs, the
position of each extra point on the centre line must be known. These distances are recorded by
measurement with a tape, the tape being positioned horizontally between appropriate centre
line pegs. The method of booking longitudinal sections should always be by the height of
collimation method since many intermediate sights will be taken. Distances denoting chainage
should be recorded for each level and most commercially available level books have a special
column for this purpose. Careful booking is required to ensure that each level is entered in the
level book with the correct chainage. Good use should be made of the 'remarks' column in this
type of levelling so that each point can be clearly identified when plotting.
6.3.2 Cross sections
A longitudinal section provides information only along the centre line of a proposed project. For
works such as sewers or pipelines, which usually are only of a narrow extent in the form of a
trench cut along the surveyed centre line, a longitudinal section provides sufficient data for the
construction to be planned and carried out. However, in the construction of other projects such
as roads and railways, existing ground level information at right angles to the centre line is
required. Taking cross sections provides this. These are sections taken at right angles to the
centre line such that information is obtained over the full width of the proposed construction.
For the best possible accuracy in sectioning a cross-section should be taken at every point
levelled on the longitudinal section. Since this would involve a considerable amount of
fieldwork, this rule is generally not observed and cross-sections are, instead, taken at regular
90
intervals along the centre line usually where pegs have been established. A right angle is set
out at each cross-section either by eye for short lengths or by theodolite for long distances or
where greater accuracy is needed. A ranging rod is placed on either side of the centre line to
mark each cross-section.
The longitudinal section and the cross-sections are usually levelled in the same operation.
Starting at a temporal benchmark (TBM), levels are taken at each centre line peg and at
intervals along each cross-section. These intervals may be regular, for example, 6m, 20 m, 30
m on either side of the centre line peg or, where the ground is undulating, levels should be
taken at all changes of slope such that a good representation of existing ground level is
obtained over the full width of the construction. The process is continued taking both
longitudinal and cross-section levels in the one operation and the levelling is finally closed on
another known point. Such a line of levels can be very long and can involve many staff
readings and it is possible for errors to occur at stages in the procedure. The result is that if a
large miss-closure is found all the levelling will have to be repeated, often a soul destroying
task. Therefore, to provide regular checks on the levelling it is good practice to include points
of known height such as traverse stations at regular intervals in the line of levels and then, if a
large discrepancy is found.
6.3.3 Drawing cross-sections and profiles
A map view looks at the surface of the earth from overhead. Contour maps use the contour
lines to represent the third dimension of elevation. Another view of the earth is a profile view.
A profile or cross section shows a cut through the earth. The top line on the cross section or
profile represents the surface of the earth. We can apply this alteration of perspective to
contour maps. From the information provided by the contour map, we can produce a cut across
this surface into the earth and, thus, show a side view like a silhouette or skyline. Specifically,
this illustration is called a topographic profile.
A topographic profile is a diagram that shows the change of elevation of the land surface along
a given line. As indicated above, it represents graphically the skyline viewed from a distance.
The vertical scale is the scale used to plot the elevation. It is usually larger than the horizontal
or map scale, exaggerated, in order to emphasize the difference in the relief. The maximum
relief is the difference in elevation between the highest and lowest points.
91
Following are the steps for drawing a topographic profile:

Lay the edge of a strip of paper along the line between the starting and ending points for
the profile.

Mark on the edge of the strip the EXACT places where each CONTOUR, STREAM, and
HILLTOP crosses this line.

Label these marks with the elevation and correct identification.

Mark any important other features such as bottoms of depressions or landmarks to be
included.

If a graph is not provided, construct the horizontal line for your profile of the SAME LENGTH
as your profile (unless a different horizontal scale is to be used for the profile.) Generally,
the same horizontal scale is used. Prepare the VERTICAL SCALE by lightly drawing lines
parallel to your horizontal base line on the proper scale for each of the elevations to be
represented. Label these lines with the correct elevations starting one or two intervals
below the lowest elevation that will be plotted (lowest elevation on the profile). Thus, the
side represents a kind of graphic scale.

Place the edge of the strip of paper with the labelled contour lines at the bottom of the
profile base line and project each contour and feature to the horizontal line of the same
elevation. Put a small dot at the intersection of these two lines.

Connect all of the points with a smooth line being careful to show all hilltops at the proper
height and all valleys and depressions at their correct approximate values.
92
Figure 6. 6: Drawing a topographic profile
6.4 Questions

List the differences between the direct method and the indirect method.

Indicate also where a method more applicable than the other.

Briefly describe the graphical interpolation of contours.
93
94
KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY, KUMASI
COLLEGE OF ENGINEERING
B. Sc. (Civil & Geological Engineering) Second Semester Examination, 2008
Second Year
May, 2008
GE 282 PRINCIPLES OF LAND SURVEYING
Three (3) Hours
SECTION A
1. Surveys covering areas so large that the spherical or spheroidal shape of the earth has to be taken
into account are called . . . . . . .
(a) Mining surveys
(b) Geodetic surveys
(c) Plane surveys
(d) Engineering surveys
2. . . . . . . are carried out for the production of plans showing property boundaries and sizes of tracts of
land required for assessment of properties and computation of land taxes.
(a) Cadastral surveys
(b) Construction surveys
(c) Hydrographic surveys
(d) Topographic surveys
3. Which of the following statements is not correct?
(a) An error is the difference between a measured value for a quantity and its true value.
(b) The exact error present in a measurement is always unknown.
(c) Systematic errors are associated with the skill and vigilance of the surveyor.
(d) Small errors occur more frequently than large ones.
95
4. Discrepancy denotes
(a) the absolute nearness of measured quantities to their true values.
(b) the difference between two measured values of the same quantity.
(c) the degree of consistency of a group of measurements.
(d) the spread of the measured values of a quantity.
5. An Abney level is a hand-held device for measuring
I.
slope angles.
II.
bearings.
III.
horizontal angles.
IV.
vertical angles.
Which of the above is/are correct?
(a)
I and II only.
(b)
II and III only.
(c)
I and IV only.
(d)
All of the above
A distance of 159.400 m was measured between two points along a slope of 1 in 5.
Use this information to answer questions 6 – 8.
6. The slope angle is
(a)
0.2º
(b)
11.3º
96
(c)
0.98º
(d)
0.0035º
7. The horizontal distance between the two points is
(a) 31.880 m.
(b) 165.305 m.
(c) 156.305 m.
(d) 131.880 m.
8. The difference in elevation between the two points is
(a) 31.880 m.
(b) 31.808 m.
(c) 128.139 m.
(d) 31.261 m.
A line AB is measured in three sections AX, XY, and YB using a fiberglass tape of nominal length 20 m.
The tape was read to the nearest 0.01 m and distances obtained for the three sections AX, XY, and YB
were 79.45 m, 8.70 m, and 126.35 m, respectively.
There is a constant slope from A to X and from Y to B, and stepping was carried out between X and Y
due to very steep ground. The reduced levels of points A, X, Y, and B are 37.62 m, 32.14 m, 19.47 m,
and 20.21 m, respectively. Before measurement, the tape was measured against a reference steel tape
and found to be 20.015 m.
Use this information to answer questions 9 – 15.
9. The standardization correction to be applied to measured length of section AX is
(a) - 0.060 m.
97
(b) + 0.060 m.
(c) - 0.015 m.
(d) + 0.015 m.
10. The slope correction to be applied to the measured length of section AX is
(a) – 0.189 m.
(b) + 0.189 m.
(c) – 0.378 m.
(d) + 0.378 m.
11. The horizontal length of section AX is
(a) 79.699 m.
(b) 79.390 m.
(c) 79.201 m.
(d) 79.321 m.
12. The slope correction to be applied to the measured length of section XY is
(a) zero because the ground between that section is very steep.
(b) zero because stepping was used.
(c) – 9.226 m.
(d) + 9.226 m.
13. The horizontal length of section XY is
98
(a) 8.693 m.
(b) 17.926 m.
(c) 17.933 m.
(d) 8.707 m.
14. The standardization correction to be applied to measured length of section YB is
(a) - 0.095 m.
(b) + 0.095 m.
(c) - 0.002 m.
(d) + 0.002 m.
15. The horizontal length of section YB is
(a) 126.443 m.
(b) 126.253 m.
(c) 126.447 m.
(d) 126.257 m.
16. A plot of land measures 45 cm 2 on a plan of scale 1:2500. What is the equivalent area of the plot on
the ground?
(a) 28,125.00 m2
(b) 281.250 m2
(c) 28.125 m2
(d) 112.50 m2
17. A triangular plot measures 60 m by 50 m by 36 m on the ground. When plotted on a plan of scale
1:P, its equivalent area on the plan is 143.786 mm 2. What is the value of P?
99
(a) 5,000
(b) 500
(c) 2,500
(d) 250
18. A zenith angle is an angle
(a) between 0 and 180º, measured in a vertical plane downward from an upward directed vertical
line through the instrument.
(b) between 0 and 90º, measured in a vertical plane downward from an upward directed vertical line
through the instrument.
(c) between 0 and 90º, measured in a vertical plane from a horizontal line upward or downward.
(d) between 0 and 360º, measured in a horizontal plane from a given reference line.
19. In a traverse survey, if the stations are occupied in a clockwise direction,
(a) the interior angles are measured clockwise in a horizontal plane.
(b) the interior angles are measured anticlockwise in a horizontal plane.
(c) the interior angles are measured anticlockwise in a vertical plane.
(d) the interior angles are measured clockwise in a vertical plane.
20. The horizontal angle measured clockwise from the true meridian to a line is
(a) the true magnetic bearing of the line.
(b) the true meridian of the line.
(c) the true bearing of the line.
(d) the true declination of the line.
21. Find the quadrantal bearing of a line whose whole circle bearing is 157° 46′ 25″.
(a) N 12° 13′ 35″ W
(b) S 22° 13′ 35″ E
100
(c) S 22° 46′ 25″ W
(d) N 12° 46′ 25″ E
22. The bearing of a line AB is 175° 45′ 27″. What is the back-bearing of the line BA?
(a) 355° 45′ 27″
(b) - 4° 14′ 33″
(c) 175° 45′ 27″
(d) 175° 45′ 27″  180°
.
23. The left-hand angle at a traverse station is
(a) the horizontal angle measured clockwise from the back station to the forward station.
(b) the vertical angle measured clockwise from the back station to the forward station.
(c) the horizontal angle measured clockwise from the forward station to the back station.
(d) the horizontal angle measured anticlockwise from the back station to the forward station.
24. The bearings of two lines PQ and PT are 060° 45′ 27″ and 290° 30′ 42″ respectively. Calculate the
clockwise angle TPQ.
(a) 229° 45′ 15″
(b) 130° 14′ 45″
(c) 069° 29′ 18″
(d) 351° 16′ 09″
25. Magnetic declination is defined as
(a) the horizontal angle from the grid north to the magnetic north.
(b) the horizontal angle from the true meridian to the magnetic meridian.
(c) the angle of inclination of the magnetic needle to the horizontal.
101
(d) the horizontal angle from the true north to the grid north.
26. The true bearing of a line is given as 169° 29′ 18″. This line is to be traced by using a compass when
the magnetic declination is 6° 35′ W. What should be the reading on the compass to observe this line?
(a) 169° 29′ 18″
(b) 169° 35′ 53″
(c) 162° 54′ 18″
(d) 176° 04′ 18″
27. A change point, as used in levelling, is
(a) a staff position where a backsight is taken followed by a foresight.
(b) an instrument position where a backsight is taken followed by a foresight.
(c) a staff position where a foresight is taken followed by a backsight.
(d) an instrument position where a foresight is taken followed by a backsight.
Table 1 below shows a Surveyor’s spirit levelling bookings (all values recorded in metres). After exposure
to rain, some of the entries have become illegible as indicated by the letters X, W, T, Q, Y, and K. Use
the information in Table 1 to answer questions 28 – 38.
Table 1:
Backsight
Intermediate
sight
Foresight
1.612
Height of
Initial Reduced
Collimation
Level
X
32.110
BM A(32.110 m)
31.400
Peg 1
32.007
Peg 2
2.312
W
1.715
34.098
Remarks
102
1.862
T
Peg 3
Q
Peg 4
Y
32.305
Peg 5
1.859
32.208
Peg 6
31.397
BM B (31.400 m)
1.957
1.988
34.067
K
28. How many change points were used in all?
(a) 10
(b) 5
(c) 3
(d) 2
29. What is the value of X?
(a) 30.498 m
(b) 33.722 m
(c) 34.422 m
(d) 32.110 m
30. Which of the following values is equal to W?
(a) 0.376 m
(b) 2.901 m
(c) 1.715 m
(d) 2.091 m
103
31. At which instrument station was the greatest number of staff readings taken?
(a) First
(b) Second
(c) Third
(d) Fourth
32. Which of the following sets of readings were taken from the same instrument position?
(a) 1.612, 2.312, 1.715, W
(b) 1.612, W, 1.957
(c) W, 1.862, 1.988
(d) 2.312, 1.862, Y, 1.859
33. What is the value of T?
(a) 32.236 m
(b) 32.110 m
(c) 35.960 m
(d) 33.869 m
34. The values of Y and Q are respectively
(a) 32.110 m, 1.672 m.
(b) 1.762 m, 32.110 m
(c) 1.672 m, 32.110 m
(d) 32.110 m, 0.195 m
35. Find the value of K.
104
(a) 2.676 m
(b) 2.760 m
(c) 2.070m
(d) 2.670 m
36. What is the misclosure of the levelling task?
(a) – 0.003 m
(b) +0.003 m
(c) +0.009 m
(d) -0.009 m
37. Calculate the correction to be applied to the initial reduced level of Peg 5.
(a) - 0.003 m
(b) +0.002 m
(c) - 0.002 m
(d) +0.003 m
38. The adjusted reduced level of Peg 6 is
(a) 32.208 m
(b) 32.210 m
(c) 32.211 m
(d) 32.205 m
The coordinates of two traverse stations A and B are given in Table 2 below.
Table 2
105
Station
Northing (m)
Easting (m)
M
894.25
450.78
P
674.85
735.68
Use the information in Table 2 to answer questions 39 – 41.
39. What is the latitude of line MP?
(a)
219.40 m
(b)
284.90 m
(c) – 219.40 m
(d) – 284.90 m
40. What is the departure of line PM?
(a)
284.90 m
(b) – 284.90 m
(c)
219.40 m
(d) – 219.40 m
41. What is the bearing of line PM?
(a) 52° 24′ 01″
(b) 232° 24′ 01″
(c) 322° 24′ 01″
(d) 307° 35′ 59″
Table 3 below shows an incomplete computation of coordinates of traverse stations for closed loop
traverse ABCDEA. Moreover, after exposure to rain, some of the entries have become illegible as
indicated by the letters Z, R, and H. Given the northing and easting coordinates of station A as 1500.00
m and 2000.00 m respectively, use the table to answer questions 42 – 49.
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From
Bearing
Length (m)
Latitude (m)
Departure(m)
Northing(m)
Easting(m)
To
1500.00
2000.00
A
A
218° 21′ 00″
Z
-122.50
-96.92
B
B
142° 48′ 00″
140.50
R
+84.95
C
C
28° 34′ 30″
150.00
+131.73
+71.75
D
116.50
+101.27
-57.58
A
D
H
42. What is the value of Z, i.e. the length of line AB?
(a) 24399.74 m
(b) 256.20 m
(c) 156.20 m
(d) - 219.42 m
43. Calculate the latitude of line BC corresponding to the value of R.
(a) – 197.57 m
(b) – 111.91 m
(c) – 191.57 m
(d) +84.95 m
44. What is the bearing of the traverse leg DA?
(a) 29° 37′ 20″
(b) 330° 22′ 40″
(c) 150° 22′ 40″
(d) 209° 37′ 20″
45. Find the linear misclosure of the traverse.
(a) - 1.410 m
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(b) 2.190 m
(c) 0.780 m
(d) 2.605 m
46. What is the fractional misclosure of the traverse corrected to the nearest hundred?
(a) 1 : 8,000
(b) 1 : 2,000
(c) 1 : 200
(d) 1 : 14,700
47. What are the adjusted coordinates of station B?
(a) 1377.89 mN and 1902.48 mE
(b) 1902.48 mN and 1377.89 mE
(c) 1377.89 mN and 1901.79 mE
(d) 1901.97 mN and 1377.89 mE
48. Compute the adjusted coordinates of station C.
(a) 2058.87 mN and 1266.33 mE
(b) 1266.33 mN and 2058.87 mE
(c) 1266.33 mN and 1986.87 mE
(d) 1986.87 mN and 1266.33 mE
49. The adjusted coordinates of station D are
(a) 1398.44 mN and 2058.04 mE.
(b) 2058.04 mN and 1398.44 mE.
(c) 71.16 mN and 132.10 mE.
(d) 132.10 mN and 71.16 mE.
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50. The coordinates of a triangular parcel ABC are given by A(50mN,200mE), B(120mN,150mE), and
D(300mN, 175mE). Compute the area of the parcel.
(a) 32,350 m2
(b) 5,625 m2
(c) 16,175 m2
(d) 64,700 m2
ANSWER ALL QUESTIONS
SECTION B
1. (a) State Simpson’s rule for determining areas of irregular figures and give three (3) conditions
under which the formula may be used.
(b) Figure 1 below shows a parcel of land consisting of a regular section ABCD (with indicated
dimensions) and an irregular portion ADE. Side AB is perpendicular to side BC, and AD is parallel
to BC.
E
A
D
1
3
7
7
B
F
i
C
1
1
Offsets taken from AD to the irregular boundary are as follows:
Chainage (m)
0
20
40
60
80
100
120
137.2
Offset (m)
0
5.49
9.14
8.53
9.75
10.76
12.50
0
By using the Simpson’s rule, compute the area of the parcel in hectares.
109
2. (a) Define contours and state three (3) characteristics of contours.
(b) Draw a 3 cm grid of the data below and plot the 76 th and above contours at 2 m vertical interval.
3.
7
4
7
4
7
7
7
8
9
0
7
6
7
5
8
6
8
5
7
5
7
5
9
0
9
0
7
6
7
9
8
5
8
6
8
5
7
5
7
4
7
5
7
6
7
5
7
4
1
0
(a) Briefly explain what is meant by stadia tacheometry.
(b) During stadia tacheometry work in a detail survey of KNUST campus, a theodolite having a
multiplying constant of 100 and an additive constant of 0 was correctly centred and levelled at
a height of 1.490 m above a traverse station Q of reduced level 46.870 m. A levelling staff was
held vertically at a traverse station P to provide a reference direction and then at the bases of
two electric poles labelled D and L in turn. The readings shown in Table 4 were taken.
Table 4
Staff position
P
Staff reading (m)
-
Vertical circle reading
Horizontal circle reading
-
000° 00' 00"
Electric pole D
1.981, 1.497, 1.013
90° 08' 20"
169° 33' 45"
Electric pole L
1.773, 1.456, 1.142
88° 13' 00"
275° 12' 25"
Given that the coordinates of station Q are 721.33 m E, 619.47 m N, and the whole circle bearing
of the line PQ is 218° 12' 20", compute:
110
(v)
the reduced levels at the bases of electric poles D and L;
(vi)
the horizontal distances of lines QD and QL;
(vii)
the bearings of lines QD and QL;
(viii)
the latitudes and departures of QD and QL;
(ix)
the coordinates of the bases of electric poles D and L.
Dr. Isaac Dadzie
Dr. E. M. Osei Jnr.
111