Leveling

Introduction
This chapter describes the various heighting procedures used to obtain the elevation of points of
interest above or below a reference datum. The most commonly used reference datum is mean
sea level (MSL). There is no such thing as a common global MSL, as it varies from place to
place depending on local conditions. It is important therefore that MSL is clearly defined
wherever it is used.

The engineer is, in the main, more concerned with the relative height of one point above or
below another, in order to ascertain the difference in height of the two points, rather than a direct
relationship to MSL. It is not unusual, therefore, on small local schemes, to adopt a purely
arbitrary reference datum. This could take the form of a permanent, stable position or mark,
allocated such a value that the level of any point on the site would not be negative. For example,
if the reference mark was allocated a value of 0.000 m, then a ground point 10 m lower would
have a negative value, minus 10.000 m. However, if the reference value was 100.000 m, then the
level of the ground point in question would be 90.000 m. As minus signs in front of a number
can be misinterpreted, erased or simply forgotten about, they should, wherever possible, be
avoided.

The vertical height of a point above or below a reference datum is referred to as the reduced level
or simply the level of a point. Reduced levels are used in practically all aspects of construction:
to produce ground contours on a plan; to enable the optimum design of road, railway or canal
gradients; to facilitate ground modelling for accurate volumetric calculations. Indeed, there is
scarcely any aspect of construction that is not dependent on the relative levels of ground points.

Leveling is the most widely used method for obtaining the elevations of ground points relative to
a reference datum and is usually carried out as a separate procedure from that used for fixing
planimetric position. Leveling involves the measurement of vertical distance relative to a
horizontal line of sight. Hence it requires a graduated staff for the vertical measurements and an
instrument that will provide a horizontal line of sight.

Definition of basic terms

1. Datum:- A datum is any reference surface to which the elevation (vertical distance) of a
points are referred. The most commonly used datum is that of mean sea level.
2. Elevation: - Elevation is the vertical distance of a points above or below on assumed
datum (level surface).
3. Leveling:- The process or methods of determining the vertical distance of a points
relative to on assumed level surface.
4. Level line:- is the surface of which it has a constant height relative to mean sea level.
5. Horizontal line:- this is a line which is tangential to the level line or a line which is
normal to direction of gravitas
 Figure 5.1 Horizontal and level lines

6. Bench Mark (BM):- are permanent reference points or marks at which their elevation
(reduced level) has been accurately determined by leveling from other permanent BM.
7. Reduced level (RL):- is the height above or below a reference datum- similar to
elevation.
8. Temporary bench mark (TBM):- are marks let up on stable points near construction sites
which all leveling operation on that particular site will be referred.
9. Back sight (BS):- is the staff reading taken on points of known elevation as a BM or a
turning points.
10. Fore Sights (FS):- is the staff reading on points whose elevation is to be determined as a
turning points. It is the last staff reading denoting the shifting of the instruments.
11. Intermediate sights (IS):- any other staff reading taken on a points at unknown elevation
from the same set up of the level. All sights b/n BS & FS are IS.
12. Turning points (TP):- is a point denoting the shifting at the level. It is the point on which
the back a fore sight are taken.
13. Station:- is a points of which whose elevation is to be determined.
14. Height of instruments:- is the elevation of plane of collimation (plane of sight) where the
instruments is correctly leveled.

Equipments used in leveling

Basically three equipments are needed.
1. Level – to give the true horizontal line
2. Staff – to read vertical height
3. Tape – to measure height of instruments
Note: There are three types of level
 1. tilting
2. automatic
3. digital

Figure 5.2 Leveling staff

Figure 5.3 Tilting level

Principle of leveling
The instruments are set up and correctly leveling in order to make the line of sight through the
telescope horizontal. If the telescope is turned through 3600 a horizontal plane of sight is swept
out vertical measurements from this plane, using graduated leveling staff enable the relative
portion of the ground points to be ascertained. Consider fig below.

Line of Collimation 1.00m

3.00m Fore sight (FS)

Back sight BS B

A
100,000m

With the instruments set up approximately midway between ground points A & B. If the
reduced level (RL) of points A is known and equals to 100,000m above a certain reference
datum then the reading at 3.00m on vertically herd staff at A gives the reduced level of
horizontal line of sights as 103,000m. This sights on to A is termed as back sights (BS) and
reduced level of the line of sights is called height of plane at collimations (HPC)
Thus,
RLA + BS = HPC . . .
The reading of 1,000m on to staff a B is called foresight (FS) and shows the ground point B to be
1,000 below HPC therefore its RL = (103,000 – 1,000)
= 102,000m
Then this is the basic concept of leveling which is then developed in to following leveling.

FS (BS) IS FS
IS
BS
2.0 2.5 3.0

1.50 2.5 0.50

B
E
C
TBM D A F
G

D (IS)
C (TBM)
F (IS)
E (FS, BS)
G (FS)
A

Let RL be reduced level

R = Staff reading.
Then
RLC = TBM
RLD = RLC + (RC – RD)
RLE = RLC + (RC – RE)
RLF = RLE + (RE – RF)
RLG = RLE + (RE – RG)

Rise and fall

The basic concept of rise and fall is illustrated by the above fig.
Let RLC = TRM = 100, 000m above BM
The line of sight from the instruments at A is truly horizontal. It can be seen that the higher
reading of D i.e 2.50 indicates that it is lower than C (TBM).
This can be written
1.5 – 2.50 = -1.0 indicated fall from C to D
Similarly from C to E
1.5 – 0.5 = +1.0 indicating the rise from C to E.
If the reduced level of TBM = RLC then
RLD = RLC + (RC – RD) but RC – RD = fall
  = 100.00 – 1 = 99.00
RLD = RLC + fall

RLE = RLC + (RC – RE)

But RC – RE = Rise
 = 100 + 1 = 101

RLE = RLC + rise

* This method of reducing staff reading gives system of booking known as rise and fall method.

Method of booking
There are two methods of booking in the field for leveling.
1. Rise & fall method
2. Height of collimation method.

Method – 1 Rise and fall methods

The readings are booked in a level book which is specially printed for the purpose as shown in
the following table

Staff Position BS IS FS Rise Fall RL Remark

The reductions of these readings are carried out in the same book. Each reading entered on
different line in the applicable column except where the points, where a foresight and back sight
occupy the same line.

Note The very important check must be applied to the reductions.

 BS   FS   Rise   Fall  Last RL  First RL

It follows from the above that the first two check should be carried out and verified before
working out the reduced level.

Method – 2 Height of plane of collimation method

The height of plane of collimation methods sometimes called height of instruments. The height
of collimation is obtained by adding the staff reading, which must be back sight, to know RL of
the points on which the staff stands. All other reading are deducted from the height of
collimation until the instruments setting is changed. Where upon the new height of collimation is
determined by adding the back sight to the RL of the change points.
The reading and Computed values are booked is a level books which is specifically printed for
this purpose.
Staff BS IS FS HPC(HI) RL Distance
position

Note: The arithmetic check to be applied to this system of booking are

(BS) - (FS) = Last RL – First RL

(All except the first) = (each HPC) * (No. IS and FSs deducted from it) - (FS + IS)

This second check is cumbersome and is often ignored so that as consequence, the intermediate
RL are unchecked. In this case, errors could go unchecked (compared with rise and fall method
where errors in all RLs are detected). Reduction is easier in height or collimation method (or
height of instrument method) as sometimes called leveling are taken from each position of
instruments.

The comparison of line of collimation method and Rise – fall method

Height of collimation Rise fall method

1. It is more rapid & soue time 1. It is laborious to compute the rise & fall
then RL
2. this method is use for reduction of level 2. It is well adopted for determining d/c in
for construction work elevation of two points.
Such as longitudinal or cross sectional
leveling operation.
3. There is no check for reduction of RLs 3. There is complete check on reduction of
of intermediate site RLs of IS.
4. There are only two arithmetic check. 4. There arithmetic check.
5. Errors if any in IS are got detected 5. Errors in IS are detected.

Misclosure, Limits and its distribution

Misclosure is leveling operation are an indication of the accuracy of the work. It is important to
realize the amounts of misclosure in leveling can only be assessed by
1. Connecting the leveling back to the BM from which it started or
2. Connecting in to another BM of known and ground elevation.
When the misclosure is assessed, one must then decide if it is acceptable or not.
In many cases depending upon the terrain and the kinds of work the Engineer decide based upon
the tolerance required.

Alternating the permissible may be based on the distance traveled or no. of set up involved .
A Common Criteria used to assess the misclosure (E) is
=m k

Where K = distance leveled in Km .

M = Constant in mm (usually from 2-12 mm)

In many case in Engineering the distance involved is quite short but the no. of setup is quite
high, in which case the following criteria most be used .

E=M n

Where n = No ob instrument setup

M = Constant is mm ( 5)

If the misclosure is outside the allowable then the leveling must be repeated and if it is with in
the misclosure has to be contributed equally to all set up

E
Correction per set up =
n

Mistakes & errors in Leveling

Some of the mistakes commonly made in leveling are
1 confusion of the of numbers is reaching of the staff example 2.345 2.0 3.345
2 Recording the back sight is foresight column and vice- versa.
3 Faulty addition a subtraction of back sight of foresight is checking every page between
bench marks.
4 Rods or staff not held in the same point for foresight and back sight in turning point. etc.
5 Instrumental level.
The errors in leveling might occur due to
1 Instrumental error
2 Field error.
3 Effect of curvature refraction.

1. Instrument error: - these are error which occurs due to the defects of instrument such as.
A) Collimation error-: The error occurs if the line of the sight is not truly horizontal when
the tubular bubble is centered i.e the line of sight is inclined up or down from the
horizontal. A Cheek known as Tow –peg’ test is used.

a1

e Horizontal line
Two – peg test: - On relative flat site establish tow pegs A & B about 50m apart and set up the
instruments of P. 8 points halt way between them. After careful leveling and focusing, sight on
the staff held at B and record reading a1. Repeat with the staff held at B and record reading b1.
Assume the line of collimation is not horizontal but inclined at an angle e, the collimation error
then the true difference in elevation between A & B is given by

a1 b1
d1.e e e d2.e

d1 d2
A B

hAB = (a1 – d1e) – (b1 – d2e)

Since the instrument is mid – way between A & B

d1 = d2
 hAB = a1 – b1 ………………….. 1
To check this again set up the leveling at Q of a distance of d3 (25m) form A or B.

a2
b2
(d1+d2+d3) d3.e
e

B
Q
A
(d1+d2) d3
hAB = [a2 – (d1 + d2 + d3).e] – [b2 – (d3.e)]
= (a2 – b2) – (d1 + d2).e ……………………….. 2

Equating the two equation

(a1 – b1) = (a2 – b2) – (d1 + d2).e.

Therefore collimation error

(a2  b2 )  (a1  b1 )
e
2d1

for filling level an average precision i.e collimation error should be less than 0.00005 red
(0.5mm per 10m).
If the error is greater than this the level should be adjusted with the instruments still set at a
horizontal lien of collimation would give a reading on the staff at A at
a1 – (d1 + d2 + d3)e

B) Defect of staff: - It is possible that the staff production may be incorrect and new or repaired.
The staff shall be corrected using steel tape. Particular attention shall be said to the base of
the staff to see. If this is the case then the staff has zero error. This does not affect the height
difference if the same staff is used for all leveling. But introduce error if two staff are to be
used for the same series of leveling.

C) Tripod defects: - stability of tripod should be checked before any field work. If the metal
shoes at the base of each leg are not loose once extended the leg can be tightened
insufficiently.

2. Field Error: - These are errors which occur due to the following.
1) Staff not vertical
2) Handling the instruments & tripod

A) Staff not vertical:- Since the staff is used to measure the vertical distance b/n ground and the
line of sight, failure to hold the staff vertical will result in incorrect reading.

B) Handling the instruments and tripod:- The HI may be altered for any set up if the tripod is
held or leant against while leveling. Avoid contact the tripod and only use the level by contact
via finger tips. If at any state the tripod is disturbed it will be necessary to repeat the instruments
set up and all the reading taken from that instruments position.

3. Effect of Curvature & Refractions

The two points A & B at exactly the same level. An instruments set up at x would give
horizontal line of sight through x’. If graduated leveling staff is held vertically on A the
horizontal line would give reading. A’. Theoretically, as B’ is at the same level do A the staff
reading should be identical (B’) but due to horizontal line reading is B’’ (ignoring refraction).
Subtracting the vertical height AA’ from BB’’ indicates that point B is lower than point A by an
amount B’B’’.

The error (c) is caused by the curvature of the earth and its value may be calculated as follows.
Taking x’ B’’ O
D

Horizontal line
Refracted ray
B’’
A’
X’
y
X
A B’ Error due to
B curvature = CC
e
R

Direction Level line

of gravity

O R

(x’B’’)2 = (O’B’’)2 – (Ox’)2

D2 = (R + C)2 – (R)2
D2 = R2 + 2RCc + C2c – R2
 D2 = Cc(2R + Cc)
 Cc = (exact).
But Cc is negligible compared to R then

If the distance D is in kilometer and radius of the earth is assumed to be 6370km

Cc = 0.0785 D2 in meter

In practice the staff reading would not be B’ but at y due to refraction or the line of sight through
the atmosphere. In general it is considered that the effect is to bend the line of sight dawn,
reducing the effect of curvature by 1/7th.
Thus the combined effect of curvature & refraction

Cc = 0.0673 D2 in meter
Reciprocal leveling: - By means of reciprocal leveling, the need for applying the above
correction may be avoided. When it is necessary to carry leveling across a linear or any
obstacles requiring long sight between two points so situated that no place to level can be found
from where the length of foresight and back sight will be even approximately equal, a special
method i.e reciprocal leveling must be used to obtain accuracy and to eliminate the following.
1. Error in instrument adjustment
2. Combined effect of earth’s curvature and refraction of atmosphere.
The level set at a point near A and staff reading are taken on A and B with bubble in center of its
ron since BM A is very near to the instruments no error due to curvature, refractions and
collimation will be introduced in the staff reading of A but there will be on error e (or a
collimation) is the staff reading at B.

The level is then shifted to the other bank on a point very near BM, B and the reading are taken
on staff held at B and A. Since B is very near, there will be no error due to three factors is
reading the staff but the staff reading on A will have on error e.
Let – a1 and b1 be corresponding staff reading at A and B when the level is get at A
– a2 and b2, be corresponding staff reading at A and B when the level is set at B.
= error due to c – r & collimation.
Horizontal line
a1

Line of sight e
A b1
Level line
B

River

hAB = a1 – (b1 – e) ……………………………………………. 1

Horizontal line
b2

e
a2
Level line Line of sight
A

hAB = (a2 – e) – b2 ……………………………………………. 2

Taking the average of the two true difference in elevation

2hAB = [a1 – (b1 – e) + (a2 – e) – b2]

hAB =1/2(a1 – b1) + (a2 – b2)

The true difference in elevations, therefore equal to the mean of two apparent differences is
elevations obtained by reciprocal observation.

Inverted staff reading:- Reduced levels of underside of structures (bridge softest) are
determined by using staff in an inverted position , the inverted staff reading is booked in a
relevant column of the level book with negative sign , so that when reading this reading from
height of collimation of the level we get.

RA RB HPC

A ( TBM)

RLA = TBM
HPC = RLA + RA
RLB = HPC – (-RB)
= HPC + RB

Fly leveling: - The permanent bench mark can be located far away from starting points of
proposed road. So, fly leveling should be done to connect the BM with starting points of the
work in order to locates its RL and then calculate RLs of different points along the alignments.
Note: In fly leveling only the back sight and foresight reading should be recorded.

Trigonometric Leveling
Trigonometric leveling is a process of determining the differences of elevations of stations from
observed vertical angles and known distances, which are assumed to be either horizontal or
geodetic lengths at mean sea level. The vertical angle may be measured by means of an accurate
thodolite and the horizontal distances may be measured in the case of plane or geodetic
surveying.
In order to get the difference in elevation between the instrument station and the object under
observation, we shall consider the following cases:
Case 1: Base of the object accessible
Case 2: Base of the object inaccessible: instrument stations in the same vertical plane as the
elevated object
Case 3: Base of the object inaccessible: instrument stations not in the same vertical plane as the
elevated object

1) Base of the object accessible

Let it be assumed that the horizontal distances between the instrument and the object can be
measured accurately.
P = instrument station
Q = point to be observed
A = center of the instrument
D = horizontal distance between P and Q
h’ = height of the instrument
h = QQ’
S = reading of staff kept at B. M. with line of sight horizontal
 = angle of elevation from A to Q
R. L. of Q = R. L. of B. M. + S + Dtan

2) Base of the object inaccessible: the instrument stations in the same vertical plane as
the elevated object

Case A: Instrument axis at the same level

R.L. of Q = R. L. of B.M. + S + h

Case B: Instrument axis at different level

Case C: Instrument axes at very different level

3) Base of the object inaccessible: the instrument stations not in the same vertical plane
as the elevated object