LEVELLING

Introduction:

The relative position of a point in terms of vertical distance, above or below a reference surface is designated by its elevation. The reference surface is known as datum having zero elevation. Thus, elevation is considered as third coordinate, it is positive if it is above and negative if the point is below the datum. Usually sea-level is considered to be the standard datum, but some time an arbitrary assumed surface is taken as the reference. The value of MSL (Mean Seal Level) as datum is obtained by taking mean of elevations of high and low tides at several points for a long period of nineteen (19) years (Since the moon takes 19 years to reach the same position). However, any surface may be assumed as datum when relative elevation over a limited area is to be established. This is done by assigning an elevation to some permanent control point and determining the elevation of the desired points in the area with respect to the assigned value to the control point. The assigned value can be of any magnitude. For convenience a sufficiently large value should be used so that no point in the coordinate system has a negative value.

Mean Sea Level as standard DATUM:

It is a standard datum ascertained by prolonged observation in different parts of sea coast of a country. It is universally accepted datum for heights. In India the MSL is determined at nine (09)* tidal observation stations:

1. Bombay (Maharashtra) 8. Karachi (Pakistan)2. Karwar (Karnataka) 9. False point (Calcutta)3. Visakhapatnam (AP) 10.Tuticorin (TN)4. Baypore (Orissa) 11. Enmore (TN)5. Cochin (Kerala) 12. Kakinada (AP) 6. Nagapattanam (TN) 13. Mangalore (Karnataka)7. Madras (TN) 14. Kandla (Gujarat) 15. Paradeep (Orissa)

Basic Definitions: 1. Levelling: Levelling is a branch of surveying the object of which is i) To find the elevation of given points with respect to a given or assumed datum ii) To establish points at a given elevations with respect to given or assumed datum.

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The first operation is required to enable the works to be designed while the second operation is required in the setting out of all kinds of engineering works. Leveling deals with the measurements in a vertical plane.

2. Level Surface:

A level surface is defined as curved surface which at each point is perpendicular to the direction of gravity at the point. The surface of still water is truly level surface. Any surface parallel to the mean spherical surface of the earth is therefore, a level surface.

Reference: V.A. = Vertical angle, H.L= Horizontal line, L.L = Level line M.S.S = Mean spheroidal surface.

2. Level Line:

The line laying on the level surface is known as a level line. It is, therefore, normal to the plumb line at all the points.

3. Horizontal plane:

Horizontal plane through a point is a plane tangential to the level surface at that point. It is, therefore, perpendicular to the plumb line through the point.

4. Horizontal line: It is a straight line tangential to the level line at a point. It is also perpendicular to the plumb line.

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5. Vertical line:A line perpendicular to the level line is called Vertical line. The plumb

line at any place on the level surface is called Vertical line.

6. Vertical Plane: Any plane which passes through the vertical line is called vertical plane.

7. Vertical Angle:The angle between an inclined line and the Horizontal line at a given

place in a vertical plane is called as vertical angle.

8. Datum Surface:The imaginary level surface with reference to which vertical distances of

the points (above or below) are measured is called as datum surface.

9. Reduced level: The height or depth of a point above or below the assumed datum is

called reduced level.

10. Elevation:

The elevation of a point on or near the surface of the earth is its vertical distance over or below an arbitrary assumed level surface or datum. The difference in elevation between two points is the vertical distance between the two level surfaces in which the two points lie.

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11. Mean sea level:

Mean sea level is the average height of the sea for all stages of the tides at any particular place it is derived by averaging the hourly tide heights over a long period of 19 years.

12. Bench mark:

Bench mark is a relatively permanent point of reference whose elevation with respect to some assumed datum is known, it is used either as a starting point for leveling or as a point upon which to close as a check.

Methods of Levelling: Following are the methods for determination of height

1. Spirit leveling (Differential leveling)2. Trigonometrical leveling3. Barometric leveling

Principle of spirit levelling:

If a horizontal line of sight is established between two points “A” and “B” then the difference of the readings of this line on a graduated scale, placed successively on the two points A & B will give the difference in elevation between these two points A & B.

The horizontal line of sight is established with the help of an instrument, called level. Graduated scales known as staffs are normally used to determine the altitude difference.

Let height of A = H’Height of level surface ab = H’ + Aa

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Height of B can be obtained by subtracting bB from the height of level surface ab

i. e. Ht of B = H’ +Aa – bB, where (Aa – Bb) is the difference of two staff reading.By successive readings, heights of other points can be established.

Special methods of spirit levelling:1) Differential levelling.2) Profile leveling3) Cross-sectioning4) Reciprocal levelling5) Precise levelling

1. Differential leveling:The operation of spirit leveling which is employed to determine the

elevations of a number of points, some distance apart, to establish Bench Marks (BM’s) in the area is called differential leveling. It is done regardless of the horizontal positions of points with respect to each other. It is some times called fly level.

2. Profile leveling: It is the method of direct leveling the object of which is to determine the elevations of paints at measured interval along a given line in order to obtain a profile of the surface along that line.

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3. Cross-Sectional leveling:The operation of leveling, which is carried out to provide heights on either

side of the main line at right angles, in order to determine the vertical section of the earth surface on the ground, is called Cross-sectional leveling.

4. Reciprocal leveling:

The operation of leveling in which difference in elevations between the points is determined accurately by two sets of reciprocal observations is called reciprocal leveling.

Reciprocal leveling is employed when it is not possible to setup the level between two points due to an intervening obstruction such as large water body.Procedure:

1. Let A and B be two points on either side of a river bank, whose height has to be determined.

2. Keep Staff at A & B.3. Set up the instrument (first position) near the stave A, note down the

reading a1, on stave A (for this purpose as the machine is very near to the object the reading is taken through the objective) and reading b1, on stave B. The reading a1 is assumed to be correct, the reading b1 is assumed to be erroneous (due to its long distance the effect of curvature, refraction and instrumental collimation etc.,) and let this error be ‘e’.

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4. Now, shift the instrument near to the stave B (second position) and similarly note down the reading b2 from stave B and a2 from stave A, this time b2 is assumed to be correct, the reading a2 is assumed to be erroneous (due to its long distance the effect of curvature, refraction and instrumental collimation etc.,.) obviously, the reading a2 will have an error ‘e’.

5. If ‘h’ is difference in elevation between two points A & B then from First observation h = {(b1 - e) - a1} assuming B is higher Second observation h = {b2 - (a2 - e)}From both the observations we get 2h = b1 – e – a1 + b2 – a2+ eFinally, 2h = (b1 – a1) + (b2 – a2)h = ((b1 – a1) + (b1– a2))/2

Obviously, it can be seen that the error due to curvature, collimation and refraction is eliminated.

Note: It is assumed refraction (atmospheric condition) is the same during both the observations.

We can also get an expression for the error ‘e’:-Equating the above two observations:(b1 – e) – a1 = b2 – (a2 – e);b1 – e – a1 = b2– a2 + e;b1 – a1 – b2+ a2 = 2e;Therefore, ‘e’ = ((b1 – a1) – (b2 – a2))/2

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5. Precise levelling:

Precise levelling is used for establishing bench marks with great accuracy at widely distributed points. The precise levelling differs from the ordinary levelling in the following points.

High grade levels and stadia rods are used. Length of sight is limited to 100 meters. Rod reading is taken against three horizontal hairs of the diaphragm. Two rod men are employed. The adjustments of the precise levels are tested daily.

1st order—permissible error—4mm √k or 0.003125√k2nd order--- permissible error—8.4mm√k or 0.00625√k3rd order--- permissible error—12mm√k or 0.0125√k

Clinometric Method: It is an instrument which is used to measure the height difference between two places.i.e. Tan θ= Δ h ÷horizontal distance.

Tan θ=Vertical angle by observation with clinometers.Horizontal distance = either by measuring or by scaling off from the plan /map.

Trigonometric Leveling:It is an indirect method of leveling in which the difference of elevation of

the points is determined from the observed vertical angle and measured distances. The vertical angles are measured with a transit (angle measuring instrument) and distances are measured directly or computed trigonometrically. It is commonly used in topographical works to find out the elevation of the top of the buildings, chimneys etc.

If horizontal distance between A and B is measured as S and the Vertical angle of B at A is measured as θ.

Then, Δ h (difference of ht between A &B) = S x Tan θIf Height of A = HaThen Height of B = H b + S x Tan θ

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Barometric Leveling:

Principle: Elevation of a point is inversely proportional to the atmospheric pressure at the place (weight of air column above). However, the relationship between the pressure and elevation is not constant. Sudden changes in temperature humidity and varying weather conditions affect pressure. The Barometric methods are particularly suited for work in rough country where the height precision is not desirable. Instrument used is called as BAROMETER. A modified from of barometer used to find relative elevations of points is called ALTIMETER. The height can be computed using the formula:

H = (18402.6 (log h1 – log h2) ((1 + (T1 + T2)/2))/273

The Instruments are used, one on the reference and the other as the rover. In the above equation H = Difference in elevation between the two points, h1 & h2 = Barometer readings at the tower and higher points respectively.

T1 & T = Temperature of the area at that time.

In mountainous countries, the approximate elevations of various mountain peaks can be obtained with the help of Barometer, the instrument used for measuring atmospheric pressure.

The difference in pressure between two points will give approximately the difference in height between the points.

The mercury in the Barometer falls about 2.45 cm for every 300mtr rise above MSL.

Check levelling: This is to check the height of BM from another two three BM.

Classification of Leveling:Depending upon the type of instrument used, procedure adopted for

observations and refinement of computations, the leveling is classified into the following categories: -

a) Leveling of High Precisionb) Precision Levelingc) Secondary Levelingd) Tertiary Leveling:- The Tertiary leveling is again sub-divided into

(i) Single Tertiary Leveling (ST) in which the permissible error is 0.025x√K where K is the distance in km.

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(ii) Double Tertiary Leveling (DT) in which the permissible error is 0.012x√K where K is the distance in km.

Instrument for spirit leveling:

The level is an instrument design to furnish a horizontal line of sight. It consists essentially of a bubble tube attached to a telescope. The axis of the level tube (bubble axis) and the line of collimation should be parallel to each other. The instrument is provided with leveling screws by which the bubble is centered and the line of collimation is brought into a horizontal plane. The level is used to determine accurately difference in heights between the two points on the surface of the earth.

Types of levels:

1) Dumpy level2) Wye level3) Tilting level4) Auto level 5) Digital level

Adjustments of level

Temporary adjustments:There are adjustments, which are required to be made every time, and

instrument is set up and before commencing any observations. This is done in three steps:

a. Setting up of the levelb. Levelingc. Elimination of Parallax

Permanent adjustment:a. To make the vertical axis perpendicular to Bubble axis.b. Line of collimation is parallel to bubble axis.

Temporary adjustment:a. Setting of the level: It includes fixing the instrument on the

tripod and also leveling the instrument approximately by the leg adjustment.

b. Leveling: After setting of the level, accurate leveling is done with the help of the foot screws and by using plate

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levels. The object of leveling of the instrument is to make its vertical axis truly vertical. Depending upon the number of foot screws provided, there are two methods of leveling.

Some basic definitions in leveling:

1. Axis of the telescope: The line joining the optical centers of the objective with the optical centre of the eye lens is called as axis of the telescope.

2. Line of Collimation: The line joining the optical centre of the objective lens with the centre of the diaphragm (cross wires) is called as line of collimation.

3. Bubble tube axis: The line drawn tangential to the curved surface (upper) of the level tube is known as bubble axis.

Permanent adjustment of a Level:

The adjustments are generally are of a long-standing nature. They require re-adjustment only after heavy use in the field. Further, these should be constantly checked up as good results are obtained only when all these conditions (given below) are fulfilled.

1. The axis of the level tube should be perpendicular to the vertical axis.

2. The line of collimation should be parallel to the level tube axis.

To make the line of collimation parallel to the bubble axis:

In a level we have a telescope having line of collimation and a level tube having the bubble axis. In an ideal instrument the line of collimation is parallel to the bubble axis. Due to various reasons like man-handling the instrument, changes in temperature and some minor causes, this parallelism between the line of collimation and the bubble axis is lost. i.e. the line of collimation becomes inclined to the bubble axis. It can be inclined either up-wards or down-wards. In either case the readings on the leveling staffs will be erroneous. This error on the leveling staff,

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which is caused due to the inclination of the line of collimation with the bubble axis, is called as collimation error.

Errors in Leveling:

Following three types of errors may affect the quality of leveling measurements.

1. Instrumental

2. Natural

3. Personal.

Instrumental Error:-

(a)Collimation Error : If the line of sight of telescope is not parallel to level axis

And forward and back staff is not kept at the equal distance from the instrument

then it will cause serious error.

(b) Error due to sluggish bubble: - Sluggish level is constant source of annoyance and delay as the level came to rest in wrong position. However, the error may be partially avoided by obtaining the bubble after the staff has been sighted.

(c) Staff not of standard length: - Incorrect length of staff may also cause the erroneous leveling, but the effect is cancelled when included in both back and fore sight reading. For accurate leveling, the rod graduation should be lasted and compared with standard tape.

(d) Error due to defective joint: - Defective joint causes serious error in leveling and in cumulative and hence joints of extendable rods must be checked regularly.

Natural Error: -

(a) Effect of curvature : - The curvature cause increased reading on rods. If the back and fore sights are not at equal distances it may cause considerable amount of error.

(b) Refraction : -Due to refraction of light ray, the line of sight trends downwards the decreasing the staff reading. This can be compensated by keeping the staves at equal distances.

(c) Variation in temperature : - Variation of temperature may not cause appreciable error in ordinary work but may affect the accuracy standard of precision leveling due to uneven expansion of the level tube.

(d) Settlement of Tripod : - If the tripod settles in the interval between the two readings (i) fore sight and back sight then the reading taken to the fore sight

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will be small and the reading taken to the back sight will be more and thus elevation will be always more.

(e) Wild interaction : - High shaken the instruments and thus disturbs the buubble.

Personal Error: -(a) Mistakes in manipulation : - These include mistakes

in setting up the level, imperfect focusing of eye piece and of objective, errors in centering the bubble and failure to watch it after each sight and errors due to resting the hands on tripods or telescope.

(b) Rod handling: - If the rod is not in plumb, the reading taken will be too great. The error varies directly with the magnitude of the rod reading and directly as the square of the inclination.

(c) Errors in sighting: -The error is caused when it is difficult to tell when the cross hair coincides with the centre of the target in a target rod and to determine the exact reading which the cross-hair appears to cover in the case of self reading rod.(d) Mistakes in Reading the rod :-

- Reading upwards, instead of downwards - To quit a zero, even two zeros from a reading. For example 1.28 instead of 1.028 or

1.06 Instead of 1.006(e) Mistakes in recording & computing :-

- Entering the reading with digits interchanged 1.242 instead 1.422- Entering back sights & foresights in wrong column.- Omitting entry- Ordinary arithmetical mistakes.

Example:-Find the error of reading of a level staff if the observed reading is

3.845 m at the point sighted, the staff being 15 cm off the vertical through the bottom.

Solution: -

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AB is the observed reading AC is the correct staff reading

AC = √AB² – BC² = √ (3.845)² - (0.15)² = 3.842

Curvature correction: From the definition of a level surface and a horizontal line it

is evident that a horizontal line departs from a level surface because of this curvature of the earth.

Let PQ be the curved surface of the earth and on the surface A be the instrument position, PC is the stave position. We know that the line of sight is a straight line assumed to free from the effect of refraction, the level line is a curve line. Due to the curvature of the earth the readings taken on the leveling staves held vertically are always more than what they would have been if the earth had a plane surface.

The correction for curvature is obtained by subtracting from the staff readings.

The correction of curvature is positive always.

BC is the departure from the level line. Actually the staff reading should have been taken at B where level line cuts staff, but since the level provides only the horizontal line of sight, the staff reading is taken at point C. thus , the apparent staff reading is more and therefore, the object appears to be lower than it really is.

From the figure

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OC² = OA² + AC ² (∟CAO = 90°)

Let BC = Cc = Correction for curvature AB = d = horizontal distance between A & B

OB & AO = R = radius of earth in the same unit as that of d.

(R + Cc) ² = R² + d ²

R²+Cc²+2R + Cc = R² + d ² = Cc (Cc+2R) = d ²

d ² d ² Cc = ------------- = ------- (Neglecting Cc in comparison to 2 R)

2R + Cc 2 R

Cc= 0.07849 d ² meters

Refraction correction:-

The line of sight provided by a level does not remain straight. It sets bends towards the earth due to refraction as it passes through layers of “Air of different densities”. The effect of refraction is apposite to that of the curvature. The correction, as applied to staff readings is positive.

The refraction curve is irregular because of varying atmospheric conditions, but for average conditions it is assumed to have a diameter about seven times that of the earth.

The correction of refraction Cr is therefore given by d ²

Cr = 1/7 ----- (+ ve) = 0.01121 d ² 2R Where “d” is in km. Combined correction due to curvature & refraction will be

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d ² 1 d ² 6 d ² C = ----- - -------- ------ = ---- ---- (Subtractive) 2R 7 2R 7 2R

= 0.06728 d ² where “d” is in km.

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