S2 2 leveling

1
Chapter 2 (Part 1)
Measuring Vertical Distance by Differential Leveling
(or Spirit Leveling)
Objective: To give definitions as a basis for the proper understanding of
leveling
To understand the methods used in leveling
Definitions
Level surface
A level surface is a surface that is everywhere at right angles to the direction of
gravity of the earth. For all practical purposes it may be considered to be a spherical
surface with its center at the center of the earth.
Datum
A datum may be a surface or a line to which observed heights are related.
In order to relate a series of heights to each other, they must be given relative to a
common point or plane known as a datum, which may be of two types: the ordnance
datum or an assumed datum.
(a) The ordnance datum (O.D.) is the datum to which all heights shown on
Ordnance Survey (O.S.) maps are referred. The datum line is the mean sea
level at Liverpool datum.
(b) Assumed datum is used where it is inconvenient or impossible to relate the
work in hand to the ordnance datum.
Bench-mark (B.M.)
A bench-mark is a fixed point of known height above the O.D. from which the height
above O.D. of any other point may be determined.
Temporary benchmark (T.B.M.)
A temporary bench-mark (T.B.M.) is a bench-mark set up by the surveyor for his own
use for a particular task.
T.B.M.’s should be stable, semi-permanent marks, such as a wooden peg set in
concrete, or some permanent feature of an existing building, e.g. ‘top of plinth’, front
doorstep’, and so on.

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Reduced level
A reduced level is the height of a point or object stated with reference to the selected
datum for the work in hand. It is abbreviated to R.L. Provided that the starting point
of the operations is a known or assumed R.L., then the R.L.’s of the various points of
the site can be calculated from this R.L. and the staff readings taken at the various
points.
Back-sight
Back-sight is the first sight, or reading, taken after the instrument (the level) has
been set up. The sight is taken to a point whose height is known, has been assumed, or
can be calculated. It is abbreviated to B.S. and is taken at the start of the work and at a
change point.
Foresight
A foresight is the last sight, or reading, taken during leveling operations before the
instrument is moved. It is abbreviated to F.S. and is taken a change point and at the
end of operations.
Intermediate sight
An intermediate sight is any sight, or reading, taken between a B.S. and an F.S. It is
abbreviated to I.S. and is sometimes termed an inter-sight.
Change point (or Turning Point)
A change point is an arbitrary point which enables the leveling to continue from a
new instrument position. It is often also termed a turning point and is abbreviated to
C.P. or T.P.

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The line of collimation
The line of collimation is described as the truly horizontal line of sight which passes
through the optical center of the telescope of the level. The height of this line above
the datum is termed the height of instrument or height of collimation, while the
horizontal plane swept out by this line as the telescope is revolved about its vertical
axis is known as the collimation plane or plane of collimation.

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Ordinary leveling
In leveling operations, however, the typical problem facing the surveyor is that the
height of one point above datum is known and it is required to find the R.L.’s of other
points above this datum.

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Vertical Control (Benchmark) Surveys
(i) Height of collimation method
Example: In the figure given below point A1 is known to be 250.0 m above O.D. and
the R.L.’s of points A2, A3, A4, and A5 are required to be found.
Again the level is set up on a suitable spot from which the staff can be read as it is
placed on each point in turn. The observations are made and the readings are booked.
The vertical distances measured from the ground to the line of collimation are now
3.75 m at A1, 3.0 m at A2, 2.0 m at A3, 1.5 m at A4, and 2.5 m at A5. Since the R.L. at
A1 is known, the required R.L can be calculated by either of the following two ways.
The first requirement is to establish the R.L. of the line of collimation. This is done by
adding the distance measured from the line of collimation to the ground at point A1 to
the known R.L. of point A1.
For example, 3.75 m + 250.0 m = 253.75 m
This R.L. is also known as the height of instrument.
Once an instrument has been set up, the height of instrument will remain the same for
each observation made until the instrument is moved to a new position. It follows
then, that the R.L. of the line of collimation in the example will also be 253.75 m for
each of the readings taken on points A2, A3, A4, and A5. By simply subtracting each
vertical distance from this figure, the R.L. of each point is given automatically;
i.e. 253.75 m – 3.0 m = 250.75 m R.L. at A2
253.75 m – 2.0 m = 251.75 m R.L. at A3
253.75 m – 1.5 m = 252.25 m R.L. at A4
253.75 m – 2.5 m = 251.25 m R.L. at A5
This is a simple, quick, and accurate way of calculating R.L.’s.
(ii) Rise and fall method

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The difference in height between any two points is referred to as either as rise or a fall
relative to one of the points. In the example, point A1 is nearer to the datum than point
A2, therefore the ground is rising from point A1 up to point A2. It is obvious then that
the point A2 is at a greater height above datum than point A1. The difference in height
between the two points must be added to the R.L. of point A1 in order to determine
the R.L. of point A2.
In this method, the actual height of the line of collimation has no real significance
other than being the line to which vertical distances are conveniently measured from
various points on the ground, and the method relies upon the difference in height
between successive points. Each point is considered in relation to the point
immediately preceding it and whose R.L. is either known or has just been
calculated.
The difference in height is obtained by subtracting the staff readings taken on the two
points;
e.g. A1 – A2 = 3.75 – 3.0 = 0.75 m
Since point A1 now has a R.L. of 250 m, by adding the difference 0.75 m to this figure
the R.L. of point A2 is found to be 250.75 m.
It is important to remember that the reading taken on the point whose R.L. is being
calculated, is always subtracted from the reading whose R.L. is known, or has just
been calculated, in order to determine the difference in height between the two points.
The reason for that is, if the result of the calculation which gives the difference in
height between any two points is positive then the difference is a rise; if it is negative
then the difference is a fall. Since the difference in height is always added to the
preceding known R.L. to determine the required R.L., the mathematical sign will
always guarantee a correct result, provided the arithmetic is carried out correctly.
In the example, the correct sequence of calculation is therefore,
R.L. at A1 is known = 250. 0 m
R.L. at A2 = (A1 – A2) + R.L. at A1 = (3.75 – 3.0) + 250.0 = 250.75 m
R.L. at A3 = (A2 – A3) + R.L. at A2 = (3.0 – 2.0) + 250.75 = 251.75 m
R.L. at A4 = (A3 – A4) + R.L. at A3 = (2 – 1.5) + 251.75) = 252.25 m
R.L. at A5 = (A4 – A5) + R.L. at A4 = (1.5 – 2.5) + 252.25 = 251.25 m

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Level Book
Objective: To understand bookings in leveling
Types of level book
Depending on the purpose for which the leveling is being carried out, various systems
of taking observations may be employed. No matter what system is selected, the
observations are recorded, along with the other relevant information, in a ruled level
book.
Two standard layouts are readily available, namely
a. height of collimation and
b. rise and fall.
Both types of layout can be seen in Table 1, and it can be noted that six of the
columns are the same for both books.
i. Backsight (B.S.) – In which the readings are noted as they are observed in the
field.
ii. Intermediate (I.S.)
iii. Foresight (F.S.)
iv. Reduced level (R.L.) – In which the known starting or finishing R.L. and those
R.L.’s calculated from the observations are written.
v. Distance – In which other information relevant to the operations is written.
vi. Remarks
The remaining columns highlight the difference between the two rulings.
vii. Height of collimation (H. of C.) – In which the height of the instrument is
written as it is calculated for each different
instrument position.
viii. Rise (R) – In which the rises and the falls are written as they are calculated
ix. Fall (F) from the observations of the staff.
Table 1 (a) Height of collimation method
Back
sight
Inter-
mediate
Fore
Sight
Collimation Reduced
Level
Distance Remarks
(b) Rise and fall method
Back
sight
Inter-
mediate
Fore
Sight
Rise Fall Reduced
Level
Distance Remarks

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Level-book checks (Checking for mistakes)
An arithmetical check should be applied either at the end of the operation or at the
end of each page when entries are carried forward over several pages.
The checks are as follows.
(a) Height of collimation method
The sum of each collimation height multiplied by the number of reduced levels
obtained from it is equal to the sum of all the intermediate sights, foresights, and
reduced levels excluding the first reduced level.
or ∑(BS) – ∑(FS) = Last RL – First RL
(b) Rise and fall method
The sum of the back-sights minus the sum of the foresights is equal to the sum of the
rises minus the sum of the falls, and is also equal to the first reduced level minus the
last reduced level.
∑(BS) – ∑(FS) = ∑R – ∑F = Last RL – First RL

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Example:
Height of Collimation Method
Rise and Fall Method

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Error Distribution
Objective: To understand the error distribution to have adjustment in leveling
Error distribution (or Level loop adjustment)
The allowable error of closure is a function of the length or total horizontal distance
of the leveling line or circuit.
The function is expressed as error = constant × √(distance).
The higher the order of accuracy, the smaller the constant.

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Example:
Levels are run a total distance of 12.30 km from BM10 to BM25 to set three other
benchmarks along the route of a proposed roadway construction project. The fixed
and recorded elevations of BM10 and BM25 are 345.567 and 432.321 m,
respectively. When closing the line of levels on BM25, an observed elevation of
432.286 m is recorded in the field book. Adjust the benchmark elevations.
Solution:
The error of closure for the line of levels is 432.321 m – 432.286 m = 0.035 m (or
35mm).
Assuming that this accuracy for the work is acceptable, an adjustment to the
intermediate benchmark elevations can be made as shown in table below. A typical
computation, for BM102, follows:
Distance of BM102 from BM10 = 3.51 + 2.62 = 6.13 km
Correction = 0.035 × 6.13/12.30 = 0.017 m
Adjusted elevation of BM102 = 398.435 + 0.017 = 398.452 m

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Example:
Consider in a level circuit, a simple method of distribution is to allocate the error in
proportion to the distance leveled. For instance, commence from a BM at A to
establish other BMs at B, C, D and E.
Solution:
BM Elevation, m Distance, km Correction, m Adjusted
Elevation
A 20.000 0 0 20.000*
B 28.566 1.5 - 0.005 28.561
C 35.010 2.3 - 0.007 35.003
D 30.650 3.3 - 0.010 30.640
E 22.845 5.2 - 0.016 22.829
A 20.018 5.7 - 0.018 20.000
The observed value for the BM at A, is 20.018 m compared with its known value of
20.000 m, so the mis-closure is 0.018 m (or 18 mm). The distance leveled is 5.7 km.
Considering the purpose of the work, the terrain and observational conditions, it is
decided to adopt a value for constant of 12.
Hence the acceptable mis-closure is 12 (5.7)1/2
= 29 mm, so the leveling is acceptable.

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Example:
A level circuit is shown in figure below. A survey is needed for local engineering
projects. It starts at BM 20; the elevations of new benchmarks 201, 202, and 203 were
determined; and then the level survey was looped back to BM20, the point of
commencement (the survey could have terminated at any established BM). Table
below showed the starting elevation at BM20 and for the field elevations and adjusted
elevations at BMs 201, 202, and 203.
Solution:
Allowable error is 8√(4.7) = 17 mm (permissible error)

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In many instances, a closing loop with known distances is not the method used and
each reduced level is adjusted in proportion to the cumulative number of set-ups to
that point from the start.
(1) There are four set-ups, and therefore E = 5(4)1/2
= 0.01 m. As the misclosure is
only 0.008 m, the leveling is acceptable.
(2) The correction per set-up is 0.008/4 = -0.002 m and is cumulative as shown in
table below.
BS IS FS Rise Fall R.L. Adj. Final
R.L.
Remarks
1.361
0.855
2.741
2.855
2.844
2.018
0.611
1.362
2.111
0.856
3.015
1.805
1.711
2.015
0.826
0.244
1.030
1.493
1.255
1.483
0.997
1.194
0.749
1.159
20.842
19.359
20.185
19.188
19.432
18.238
19.268
20.761
20.012
21.267
20.108
-0.002
-0.002
-0.002
-0.004
-0.004
-0.006
-0.008
-0.008
-0.008
-0.008
20.842
19.357
20.183
19.186
19.428
18.234
19.262
20.753
20.004
210259
20.100
TBM ‘A’
C.P.
C.P.
C.P.
TBM ‘B’
(20.100)
7.812 8.546
7.812
4.848 5.582
4.848
20.842
20.108
0.734 0.734 0.734 Arith.
Checked

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Reciprocal leveling
When it is necessary to run levels accurately over ravines, rivers, or other obstacles
where the BS and FS distances must necessarily be different, a procedure called
reciprocal leveling may be used. This provides another way to cancel or average out
instrumental errors as well as the effects of refraction and the earth’s curvature.
The procedure involves two instrument setups, one nearby each point. From each
instrument position, a BS on point A and an FS on point B is taken and an elevation is
computed for point B. This will result in two different elevations for B due to the
natural and instrumental errors. But by averaging the two elevations, the effects of the
errors are canceled out and the true or most probable elevation is obtained.
Example:
In leveling across a river, reciprocal leveling observations gave the following results
for staffs held vertically at X and Y from level stations A and B on each bank
respectively:
Staff reading of X from A = 1.753m
Staff reading of X from B = 2.080m
Staff reading of Y from A = 2.550m
Staff reading of Y from B = 2.895m
If the RL of X is 90.37 AOD, obtain that of Y.
Note from the staff readings that Y is lower than X.
Instrument at A. Apparent difference = 2.550 – 1.753 = 0.797m
Instrument at B. Apparent difference = 2.895 – 2.080 = 0.815m
Therefore the true difference = (0.797 + 0.815)/2 = 0.806m
Therefore RL of Y = 90.37 – 0.81= 89.56 AOD.

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Two-peg test
Objective: To understand two peg test
The collimation error of the tilting level can be checked by two-peg test.
On a relatively flat site, establish two pegs A and B about 50 meters apart and set up
the instrument at P, point halfway between them. After careful leveling and focusing,
sight on the staff held vertically at A and take reading a1. Repeating with the staff held
at B and record reading b1. Assuming the line of collimation is not horizontal but
inclined at an angle e, the collimation error, then the true difference in height between
A and B is given by
∆hAB = (a1 – d1.e) – (b1 – d2.e)
Because the instrument is midway between A and B, d1 and d2 are equal, and so
∆hAB = a1 – b1
Now move the instrument to Q, a point that extends the line AB by d3. Repeat the
observations onto a staff at A and B recording the readings a2 and b2. The line of
collimation will again be inclined to the horizontal by the angle e. In this case the true
difference in height between A and B is given by
∆hAB = [a2 – (d1 + d2 + d3).e] – [b2 – d3.e)
= (a2 – b2) – (d1 + d2).e
By equating the two measures of the height difference:
(a1 – b1) = (a2 – b2) – (d1 + d2).e
)(
)()(
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1122
dd
baba
e
+
−−−
=∴
For a tilting level of average precision, e, the collimation errors, should be less than ±
0.00005 rad (equivalent to a height error of ± 0.5 mm per 10 m).,
Note the importance of this result. Even when the leveling instrument is not in correct
adjustment, the difference in height measured between two points by a level,
equidistant from each, is the true difference in height.

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Chapter 2 (Part 2)
Leveling Principles and Equipments
Introduction
Purpose of measuring vertical distance (or leveling)
1. Leveling provides data for determining the shape of the ground and drawing
topographic maps.
2. The elevations of new facilities such as roads, structural foundations, and
pipelines can then be designed.
3. Finally, the designed facilities are laid out and marked in the field by the
construction surveyor.
4. The surveyors’ elevation marks (grade stakes) serve as reference points from
which building contractors can determine the proper slope (rate of grade) of a
road, the first-floor elevation of a building, the required cutoff elevation for
foundation piles, the invert elevation for a storm sewer and etc.
Elevation – The vertical distance of a point above or below a given reference surface
is called the elevation of the point. (The words altitude and height are
sometimes used in place of elevation)
Mean sea level (MSL) – The most commonly used reference surface for vertical
distance is mean sea level.
Running levels (or Leveling) – is a process the surveyor to determine the elevations of
points.
Principles of leveling
There are several methods for measuring vertical distance and determining the
elevations of points. Traditional methods include:
i.) Barometric leveling
ii.) Trigonometric leveling, and
iii.)Differential leveling
Barometric leveling
By using special barometers (altimeters) to measure air pressure (which decreases
with increasing elevation), the elevations of points on the Earth’s surface can be
determined to within ±1m (or ±3 ft).

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This method is useful for doing a reconnaissance survey of large areas in rough
country and for obtaining preliminary topographic data.
Trigonometric leveling
Is an indirect procedure. The vertical distances are computed from vertical angle and
horizontal or slope distance data. It is also applied for topo work over rough terrain or
other obstacles.
Differential leveling (or spirit leveling)
By far the most common leveling method, and the one most surveyors are concerned
with. It comprises a telescopic sight and a sensitive spirit bubble vial.
Leveling Instruments
1) A surveying level basically consists of a telescope and a sensitive spirit bubble
vial.
2) The spirit bubble vial can be adjusted so that, when the bubble is centered, the
line of sight through the telescope is horizontal.
3) The telescope is mounted on a vertical spindle that fits into a bearing in the
leveling head.
4) The leveling head has three leveling screws.
5) The telescope can be easily rotated about its standing axis and pointed toward
any direction of the compass.

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Type of levels
i) Automatic level
ii) Digital level
iii) Tilting level (mostly obsolete)

S2 2 leveling

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