GE 282 Large Scale Surveying Lecture Notes - Web viewTitle: GE 282 Large Scale Surveying Lecture Notes Author: Dr. Isaac Dadzie Last modified by: Dr. Isaac Dadzie Created Date: 2/23/2008

GE 282 Large Scale Surveying Lecture Notes

Syllabus:(i) Distance measurements – direct and indirect distance measurements-

taping, optical (tacheometric, substance bar), EDM;

(ii) Angular measurements;

(iii) Traverse computations (reduction of forward bearings, L-D computation, computation of coordinates), and adjustments;

(iv) Area computations and subdivision of plots;

(v) Spirit and trigonometrical levelling;

(vi) 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 (9 th 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;

3) Bannister, A. and Raymond S., 1992, Surveying, Longman Group UK Ltd, ISBN 0-582-274532;

4) Uren, J. and Price, W. F., 1994, Surveying for Engineers (2nd Ed.), Macmillan Press Ltd., London, UK, ISBN 0-333-37081-3

Lecturer: Dr Isaac DadzieGeomatic Engineering DepartmentKNUSTKumasi

1 Introduction to Surveying

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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:

(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

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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;

(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

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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.

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.

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Global Positioning System (GPS) Surveys: Positioning in which the coordinates X, Y, and Z of survey stations are determined by the reception and analysis of NAVSTAR Satellite signals.

1.5 Questions1. Distinguish between plane and geodetic surveying.2. List and discuss four main classes of land surveying.

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2 Distance MeasurementsMeasurement 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 tacheometric techniques or by using electronic distance meters (EDMs) or Total Stations.

2.1 Types of Distance MeasurementDepending 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.

A B

C

Horizontal distance (H)

Vertical distance (d)

Slope distance (L)

Fig. 2.1: Types of distance measurement

θ Inclination or Slope angle

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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 MeasurementTaping 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(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 RangingWhen 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

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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 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 GroundStep-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:

(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;

D1

D2

D3

A

BD = D1 + D2 + D3

Fig. 2.2: Step-chaining

Direction of taping is usually downhill

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(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 MeasurementsIn measuring the horizontal distance between two points on a steep slope, rather than break tape every few metres, 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

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

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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 levelTo 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.

Solution 2:Let θ be the inclination (or slope) angle

For a slope of 1 in 3,

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Plan length = slope length =

Questions1. 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 ElevationMeasurements 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.

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:Slope distance, L = 290.430 mElevation difference, d = 891.4 – 865.2 = 26.2 mFrom Pythagoras theorem,

Horizontal distance,

QuestionA 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.

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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,

, where is the error in a measurement, the measured value, and 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.

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.

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2.2.7 Classification of ErrorsErrors in measurements are of two classes: Systematic and Random.

2.2.7 (i) Systematic ErrorsA 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.

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;

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(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 is a typical feature in surveying. If a quantity is measured several times with the same instruments in the same way, it is highly unlikely that all the results will be identical, even after the effects of systematic errors have been removed. The true value of a quantity is never known, except in few isolated cases, for example, in measuring the angles of a triangle where the true value of the sum should be 180o . Generally, the mean of the repeated and ‘corrected’ measurements is accepted as the true value of the quantity.

Random Errors are associated with the skill and vigilance of the surveyor. They are introduced into each measurement mainly due to the 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 ones2. 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

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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:-

Group A: 99.01, 98.99, 99.02 and 98.98m Group B: 99.85, 99.90, 100.15 and 100.00mFrom the above definitions, the results of Group A, being closely grouped together with a small spread, constitute a more precise measurement than those of Group B. However, the results of Group B are nearer to the nominal length of the line and should therefore be the more accurate of the two.

2.2.9 Applying Corrections to Tape Measurements2.2.9(i) Incorrect Length of Tape: Standardization CorrectionAn 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

and

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where = correction to be applied to the measured length of a line to obtain the true length

= actual tape length (obtained from standardization)= nominal tape length (given by manufacturers)= measured (or recorded) length of line

= 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, = 100.000 m

Actual length, = 100.020 m

Standardization correction,

Corrected length of line,

Note (i): = amount by which a full tape length is too long or too short = 0.020 m

= number of tape lengths in the measured line = 5.6575

= 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

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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:

and

where = the temperature correction to be applied (in feet or metres) = the length of the tape actually used (in feet or metres) = the coefficient of linear expansion of the material of the tape= the tape temperature at the time of measurement= the tape temperature when it has standard length

2.2.9(iii) Tape Not Horizontal: Slope CorrectionIf the tape is assumed to be horizontal but actually is inclined, an error is introduced. The amount of this error is computed as follows:

and

where = the height difference between the ends of the line= measured slope distance= slope angle= horizontal distance

2.3 Indirect Distance MeasurementsAn 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

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Tacheometry is a surveying method used to quickly determine the horizontal distance to, and elevation of, a point. 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:(i) Horizontal distance between station P and station X, which is given

by the horizontal component of the inclined line of sight, obtained indirectly as follows:

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(ii) The vertical component of the inclined line of sight given by:

(iii) The reduced level of the staff position given by

where = multiplying constant of the telescope, usually set at 100;= stadia interval on the staff= upper stadia reading – lower stadia

reading;=centre hair reading of the staff at X;= vertical angle or inclination of line of sight to the horizontal;=height of the theodolite at P;= additive constant, usually taken to be zero (0); and= reduced level of stations P and X, respectively.

2.3.1 (ii) Subtense TacheometrySubtense 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

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The horizontal angle between the targets is measured with a theodolite set over the other

end of the line (Fig. 2.7).

Fig. 2.7: Subtense Bar Tacheometry

The horizontal distance between the theodolite and the target is computed using the relation:

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.

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

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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 InstrumentThe 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.

Fig. 2.8: Measurement principle of EDMThe 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

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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 11. 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. 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?

3. 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. Calculate the horizontal length of AB.

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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 computationThe 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:(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.

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5.2.1 (i) Areas by subdivision into simple figuresThe 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 methodThe area of a triangle whose sides are known is computed by the formula

where is the semi-perimeter and , and are the lengths of the sides of the triangle.

(b) Area of a triangle with known height

(c) Area of a triangle with known included angle

where is the angle included between sides and

Worked example 1: Compute the area of a right-angled triangle of sides 3 m, 4 m and 5 m.

a = 4 m

b = 3 m

c = 5 m

Fig. 1: Right-angled triangle

Solution to example 1

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1. Using the first formula,

2. Using the second formula,

3. Using the third formula,

(2) Area from a trapezium

where and are parallel sides and 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.

Solution to example 2:

5.2.1 (ii) Areas by offsets from straight lines

a = 4 m

b = 6 m

h = 5 m

a = 4 m

b = 6 m

h = 5 m

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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) at regular intervals (or width) are measured from the traverse line to the boundary.

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:

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.

Irregular boundary, GT

Traverse line

y1 Y2 Y3 Y5 y--- Yn-1 Yn

Offsets

w w w w w wA B

G

T

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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) at regular intervals (or width) are measured from the traverse line AB to the boundary GT.

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.

Note: Simpson’s rule can be applied to an odd number of offsets only.

Worked example 3:

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.

Irregular boundary GT divided into parabolic arcs

G T

BA

Offsets y1 y2 y3 y4 y--- yn-2 yn-1 yn

Note: If number of offsets (n) is odd, number of strips would be even

0 10

20

30

40

50

60

70

R P

Boundary

Offsets 10.0

m8.

0 m

3.0

m

9.5

m 9.2

m

7.1

m 4.5

m

2.5

m

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Solution to example 3Since there is an even number of offsets between R and P at regular intervals of 10 m, Trapezoidal rule could be used.

Worked example 4The 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 4Using Simpson’s rule,

Worked example 5State 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 8.5310.67

12.50

9.76 4.57 1.83 0

Find the area between the fence and the chain line using(a) Trapezoidal rule(b) Simpson’s rule

Solution to example 5

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(a) Using Trapezoidal rule,

(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.

It is often convenient to tabulate the working.

Offset No. OffsetSimpson’s Multiplier Product

1 0 1 02 5.49 4 21.963 9.14 2 18.284 8.53 4 34.125 10.67 2 21.346 12.50 4 50.007 9.75 2 19.508 4.57 4 18.289 1.83 1 1.83

Sum=185.31

The area between offset 1 and 9 is therefore given by

Now the area between the ninth and tenth offsets is computed

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Therefore, the total area between the chain line and the fence is 1253.70 .

5.2.2 Area by coordinatesComputation 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.

For four traverse stations numbered 1 up to 4, the area enclosed by the polygon is given by

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 100.0 200.0B 205.0 300.0C 250.0 350.0D 200.0 400.0

Calculate the area in hectares enclosed by the stations

Solution to example 6:

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Traverse Station

Northing, X (m)

Easting, Y (m)

Cross coordinatePlus ( ) Minus ( )

A 100.0 200.0B 205.0 300.0 100.0 x 300.0 =

30,000.0200.0 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

Sum = 241,750.0

Sum =226,000.0

Algebraic sum = 241,750.0 – 226,000.0 = 15,750.0

5.3 Area by measurements from maps or plansTo 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 squaresThis 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.

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5.3.2 Area by scaled lengthsThe 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 coordinatesThe 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 conveniently recorded. From the file of coordinates, the area is computed using the cross coordinate method.

5.3.4 Area by planimeterA 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. 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,

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(Figure .9.54.6). Put the tracing point on a known point on the outline and read both drums with reference to the index mark.

Figure 8.1: Area by planimeter

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 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].

5.4 Sources of error in determining areasSome 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.

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34

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

Figure 6. 1: Levelling terminolgies

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.

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MSL datum

Level line

Horizontal line at point A

AElevation of point A

(measured along vertical line through

A)

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.

Level line: A level line is a curved line in a level surface.

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.

Vertical datum: Any level surface to which elevations are referred (e.g. mean sea level).

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)

Mean Sea Level (MSL)

MSL is the average height of the sea’s surface for all stages of the tide over a considerable period.

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.

Vertical control

A series of bench marks or other points of known elevations established throughout an area.

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6.2 Curvature and refractionIt 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

Figure 6.2: Earth's curvature, (a) and (b)37

A

level line through A

horizontal line at A

mean sea level

vertical at A

C

RR

d

(a)

(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.

Figure 6. 3: Principles of levelling

Referring to figure 6.9, the direction of levelling is from A to B and to C:

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0.500m

B

A

95.400m

2.500m

datum

line of collimation

CI1I2

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) = 2.500mForesight (FS) = 0.500mIf RL at A = RLA = 95.400mThen 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.

Rise and fall method

Table 6. 1: Rise and fall method

BS IS FS Rise FallInitial RL

Adjustment Adjusted RL

Remarks

2.191         49.873 0 49.873

TBM 49.873

  2.505     0.314 49.559 +0.002 49.561 A  2.325   0.180   49.739 +0.002 49.741 B3.019  

1.496 0.829   50.568 +0.002 50.570 C (CP)

  2.513   0.506   51.074 +0.004 51.078 D1.752  

2.811   0.298 50.776 +0.004 50.780 E (CP)

    3.82   2.072 48.704 +0.006 48.710 TBM

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4 48.710   6.962

8.131 1.515 2.684 48.704  

8.131 2.684 49.873  -1.169     -1.169   -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 n th instrument station is given by

Cn = n x correction per station

These cumulative adjustments are done in Table 6.1.

Height of Collimation method.

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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.

Table 6. 2: Height of Collimation method.

BS IS FS HPCInitial RL

Adjustment

Adjusted RL Remarks

2.191     52.064 49.873 0 49.873TBM 49.873

  2.505     49.559 +0.002 49.561 A  2.325     49.739 +0.002 49.741 B3.019   1.496 53.587 50.568 +0.002 50.570 C (CP)  2.513     51.074 +0.004 51.078 D1.752   2.811 52.528 50.776 +0.004 50.780 E (CP)

    3.824   48.704 +0.006 48.710TBM 48.710

6.962 7.343 8.131106.115 300.420  

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 = 52.064mRL of A = 52.064 – 2.505 = 49.559mRL of B = 52.064 – 2.325 = 49.739mRL of A = 52.064 – 1.496 = 50.568m

For instrument station I2, HPC = 50.568 + 3.019 = 53.587mThe same procedure is continued for the other stations.

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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.

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.

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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 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.

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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 = Vertical Interval BC Horizontal Equivalent

Figure 6. 5: Gradient and profile

45

4030

20

100

ED A C

0

20

40

60

B

AC

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 pegsHere 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 tacheometric 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:

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

46

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 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.

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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:

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

48

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 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

49

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.

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.

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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.

Figure 6. 6: Drawing a topographic profile

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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.

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KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY, KUMASICOLLEGE OF ENGINEERING

B. Sc. (Civil & Geological Engineering) Second Semester Examination, 2008Second 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.

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 measuringI. slope angles.II. bearings.III. horizontal angles.IV. vertical angles.

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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

(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.

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9. The standardization correction to be applied to measured length of section AX is(a) - 0.060 m.(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 (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

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(a) 126.443 m.(b) 126.253 m.(c) 126.447 m.(d) 126.257 m.

16. A plot of land measures 45 cm2 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 mm2. What is the value of P?(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.

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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

(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.(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″

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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

Height of Collimation

Initial Reduced Level

Remarks

1.612 X 32.110 BM A(32.110 m)

2.312 31.400 Peg 1

W 1.715 34.098 32.007 Peg 2

1.862 T Peg 3

1.957 1.988 34.067 Q Peg 4

Y 32.305 Peg 5

1.859 32.208 Peg 6

K 31.397 BM B (31.400 m)

28. How many change points were used in all?(a) 10(b) 5(c) 3

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(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

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

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35. Find the value of K.(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 Station Northing (m) Easting (m)

M 894.25 450.78P 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?

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(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.

From BearingLength (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

D H 116.50 +101.27 -57.58 A

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

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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(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.

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

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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.

Offsets taken from AC to the irregular boundary are as follows:Chainage (m) 0 20 40 60 80 100 120 137Offset (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.

2. (a) Define contours and state three (3) characteristics of contours.

(b) Draw a 3 cm grid of the data below and plot the 76th and above contours at 2 m vertical interval.

A

C

D137.20 m

77.36 m

E

B 110.00 mFigure 1

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3. (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 4Staff position Staff reading (m) Vertical circle

readingHorizontal circle reading

P - - 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:(i) the reduced levels at the bases of electric poles D and L;(ii) the horizontal distances of lines QD and QL;(iii) the bearings of lines QD and QL;(iv) the latitudes and departures of QD and QL;(v) the coordinates of the bases of electric poles D and L.

74.0 74.0 78.0 90.577.0

86.7 85.5 75.475.576.2

75.1 90.8 101.5

90.2 76.8

75.385.586.885.079.0

74.5 75.4 76.0 75.4 74.1

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Dr. Isaac Dadzie Dr. E. M. Osei Jnr.

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