SURVEYING – II (10CV 44)
Unit – 3 TRIGONOMETRIC LEVELLING
Date: 26/2 to 21/3/2013
UNIT – 3
TRIGONOMETRIC LEVELLING
Syllabus:
 Determination of elevation of objects when the base is accessible and inaccessible by single plane and
double plane method,
 Distance and difference in elevation between two inaccessible objects by double plane method.
 Salient features of Total Station, Advantages of Total Station over conventional instruments, Application
of Total Station.
INTRODUCTION
 This is an indirect method of levelling.
 In this method the difference in elevation of the points is determined from the observed vertical
angles and measured distances.
 The vertical angles are measured with a transit theodolite and
 The distances are measured directly (plane surveying) or computed trigonometrically (geodetic
survey).
 Trigonometric levelling is commonly used in topographical work to find out the elevation of the
top of buildings, chimneys, church spires, and so on.
 Also, it can be used to its advantage in difficult terrains such as mountaineous areas.
 Depending upon the field conditions and the measurements that can be made with the
instruments available, there can be innumerable cases.
Determination of elevation of object when the base is accessible – the object is Vertical
It is assumed that the horizontal distance between the instrument and the object can be measured
accurately. In Fig. 1, let
B = instrument station
F = point to be observed
C = centre of the instrument
AF = vertical object
D = CE = horizontal distance between the object and instrument
h1 = height of the instrument at B
h = height FE
S = reading on the levelling staff held vertical on the Bench Mark (B.M)
 = angle of elevation of the top of the object
HARISH.T.S, Dept. of Civil Engineering
Page | 1
SURVEYING – II (10CV 44)
Unit – 3 TRIGONOMETRIC LEVELLING
From triangle CFE,
tan  = FE / CE
FE = CE tan 
h = D tan 
If the reading on the staff kept at the B.M. is ‘S’ with the line of sight horizontal,
R.L. of F = R.L. of B.M. + S + h
= R.L. of B.M. + S + D tan 
The method is usually employed when the distance between the instrument and the object is small.
However, if the distance is large, the combined corrections for curvature and refraction should also be
applied.
The combined correction for curvature and refraction is given by
C = 0.06728 D2 metres, when D is in kilometers.
R.L. of F = R.L. of B.M. + S + D tan  + C
Determination of elevation of object when the base is inaccessible – the Instrument Stations
and the Elevated Object are in the Same Vertical Plane
If the horizontal distance between the instrument and the object cannot be measured due to
obstacles etc., two instrument stations are used so that they are in the same vertical plane as the
elevated object. Fig. 2
Procedure
1) Set up the theodolite at O1 and level it accurately with respect to the altitude bubble.
2) Direct the telescope towards O2 and bisect it accurately. Clamp both the plates. Read the vertical
angle 1.
3) Transit the telescope so that the line of sight is reversed. Mark the second instrument station O2
on the ground. Measure the distance O1O2 accurately. Repeat steps (2) and (3) for both face
observations. The mean values should be adopted.
4) With the vertical vernier set to zero reading, and the altitude bubble in the centre of its run, take
the reading on the staff kept at nearby B.M.
5) Shift the instrument to O2 and set up the theodolite there. Measure the vertical angle 2 to F with
both face observations.
6) With the vertical vernier set to zero reading, and the altitude bubble in the centre of its run, take
the reading on the staff kept at the nearby B.M.
Assuming the instrument stations and the object to be in the same vertical plane, the following two
cases arise.
* Instrument axes at same level
* Instrument axes at different level
HARISH.T.S, Dept. of Civil Engineering
Page | 2
SURVEYING – II (10CV 44)
Unit – 3 TRIGONOMETRIC LEVELLING
Instrument axes at same level
In Fig. 2,
Let h = FA’
1 = angle of elevation from O1 to F
2 = angle of elevation from O2 to F
S = staff reading on B.M., taken from both O1’ and O2’, the reading being the same in both the cases.
d = horizontal distance between the two instrument stations.
D = horizontal distance between O1 and F
From triangle O1’A’F, h = D tan 1 -------------------------- (i)
From triangle O2’A’F, h = (D + d) tan 2 ------------------- (ii)
From Eqs. (i) and (ii)
D tan 1 = (D + d) tan 2
or D (tan 1 - tan 2) = d tan 2
or D =

 -

Hence, h = D tan 1 =


 -

R.L. of F = R.L. of B.M. + S + h
HARISH.T.S, Dept. of Civil Engineering
Page | 3
SURVEYING – II (10CV 44)
Unit – 3 TRIGONOMETRIC LEVELLING
Instrument axes at different level
Depending upon the terrain, three cases arise:
A. Instrument axis at O2 higher that that at O1 (Fig. 3)
h1 - h2 = A’A” = S2 – S1 = S
From triangle O1’A”F, h1 = D tan1 -------------------------- (i)
From triangle O2’A”F, h2 = (D + d) tan2 -------------------- (ii)
Subtract Eq. (ii) from Eq. (i) to get
h1 - h2 = D tan1 - (D + d) tan2
S = D tan1 – D tan2 – d tan2
= D (tan1 –tan2) - d tan2
or D (tan1 - tan2) = S + d tan2
or D =

 -

But, h1 = D tan1
or h1 =

 -


and R.L. of F = R.L. of B.M. + S1 + h1
HARISH.T.S, Dept. of Civil Engineering
Page | 4
SURVEYING – II (10CV 44)
Unit – 3 TRIGONOMETRIC LEVELLING
B. Instrument axis at O1 higher than that at O2 (Fig. 4)
h 2 – h 1 = S1 – S 2 = S
From triangle O1’A”F, h1 = D tan1 -------------------------- (i)
From triangle O2’A”F, h2 = (D + d) tan2 -------------------- (ii)
Subtract Eq. (i) from Eq. (ii) to get
h2 – h1 = (D + d) tan2 - D tan1
S = D tan2 – D tan1 + d tan2
= D (tan2 –tan1) + d tan2
or D =
 –
 -

But h1 = D tan1
or h1 =
 –
 -


Hence, R.L. of F = R.L. of B.M. + S1 + h1
HARISH.T.S, Dept. of Civil Engineering
Page | 5
SURVEYING – II (10CV 44)
Unit – 3 TRIGONOMETRIC LEVELLING
C. Instrument axes at very different levels (Fig. 5 and 6)
If the difference in elevation (S2 – S1) between the two instrument stations is too large and cannot be
measured on a staff at the B.M., then the following procedure is adopted:
1)
2)
3)
4)
Set up the instrument at O1 and measure the vertical angle at the point F (Fig. 5)
Transit the telescope and establish point O2, at a distance d from O1.
Shift the instrument to O2 and measure the vertical angle at the point F.
Observe the staff reading r on the staff at O1 (Fig. 6). Let S be the difference in level between the
two axes at O1 and O2.
 S = h 2 – h1
We know that
D=
and h1 =
 –
 -

 –
 -


Height of station O1 above the axis at O2 = h – r
= d tan - r
S = d tan  - r + h’
Hence, R.L. of F = R.L. of B.M. + S1 + S + h1
= R.L. of B.M. + S1 + (d tan  - r + h’) + h1
HARISH.T.S, Dept. of Civil Engineering
Page | 6
SURVEYING – II (10CV 44)
Unit – 3 TRIGONOMETRIC LEVELLING
Determination of elevation of object when the base is inaccessible – the Instrument Stations
and the Elevated Object are not in the Same Vertical Plane
Let P and R be the two instrument stations not in the same vertical plane as that of Q. The procedure is
as follows:
1) Set the instrument at P and level it accurately with respect to the altitude bubble. Measure the
angle of elevation 1 to Q.
2) Sight to the point R with reading on horizontal circle as zero and measure the angle RPQ1 , i.e, the
horizontal angle  at P.
3) Take a backsight s on the staff held at B.M.
4) Shift the instrument to R and measure 2 and  there.
In Fig. 7,
AQ’ = horizontal line through A
Q’ = vertical projection of Q
Thus, AQQ’ is a vertical plane
Similarly, BQQ” is a vertical plane
Q” = vertical projection of Q on a horizontal line through B
PRQ1 = horizontal plane
Q1 = vertical projection of Q
R = vertical projection of B on a horizontal plane passing through P
 and  = horizontal angles
1 and 2 = vertical angles measured at A and B respectively.
From triangle AQQ’ , QQ’ = h1 = D tan 1 --------------- (1)
From triangle PRQ1 , PQ1R = 180 - ( + ) =  - ( + )
From the sine rule,
=
=
 PQ1 = D1 =
------------------ (2)
HARISH.T.S, Dept. of Civil Engineering
–
=
Page | 7
SURVEYING – II (10CV 44)
and RQ1 = D2 =
Unit – 3 TRIGONOMETRIC LEVELLING
--------------------- (3)
Substituting the value of D in (1), we get
h1 = D1 tan 1 =
tan 
 R.L. of Q = R.L. of B.M. + s + h1
As a check, h2 = D2 tan 2 =
tan 
If a reading on B.M. is taken from B, the R.L. of Q can be known by adding h2 to R.L. of B.
HARISH.T.S, Dept. of Civil Engineering
Page | 8
SURVEYING – II (10CV 44)
Unit – 3 TRIGONOMETRIC LEVELLING
Total Station
Introduction
 A Total station is an optical instrument used as a primary scheme for modern surveying.
 It is a combination of an electronic theodolite (transit), an electronic distance meter (EDM) and
software running on an external computer known as data collector.
 When these instruments are combined and interfaced with EDMs and electronic data collectors,
they become total-stations or electronic tacheometers (ET).
 With a total-station one may determine horizontal and vertical angles together with slope
distances from the instrument to points to be surveyed.
 With the aid of trigonometry and triangulation, the angles and distances may be used to calculate
the coordinates of actual positions (X, Y, and Z or northing, easting and elevation) of surveyed
points, or the position of the instrument from known points, in absolute terms.
 These are operated using a multi-function keyboard which is connected to a microprocessor built
into the instrument.
 The microprocessor in the total-station can not only perform a variety of mathematical operations
– for example,
 averaging multiple angle measurements,
 averaging multiple distance measurements,
 calculation of rectangular coordinates,
 calculation of slope corrections,
 distances between remote points,
 remote object elevations,
 atmospheric and instrumental corrections – but in some cases, can also store
observations directly using an internal memory.
 Most modern total-station instruments measure angles by means of electro-optical scanning of
extremely precise digital bar-codes etched on rotation glass cylinders or discs within the
instrument.
 The best-quality total-stations are capable of measuring angles down to 0.5 arc-second.
 The low-cost construction-grade total-stations can generally measure angles up to 5 or 10 arcseconds.
 Measurement of distance is accomplished with a modulated microwave or infrared carrier signal,
generated by a small solid-state emitter within the instrument’s optical path, and bounced off of
the object to be measured.
 The modulation pattern in the returning signal is read and interpreted by the onboard computer
in the total-station.
 The distance is determined by emitting and receiving multiple frequencies, and determining the
integer number of wavelengths to the target for each frequency.
 Most total-stations use a purpose-built glass Porro prism as the reflector for the EDM signal, and
can measure distances out to a few kilometers, but some instruments are reflectorless, and can
measure distances to any object that is reasonably light in color, out to a few hundred metres.
 The typical total-station EDM can measure distances accurate to about 3 millimeters or 1/100 th of
a foot.
Salient Features of Total-Stations
Total stations are capable of measuring angles and distances simultaneously and combine an
electronic theodolite with a distance measuring system and a microprocessor.
Angle measurement
The axis configuration is identical and comprises the vertical axis, the tilting axis and line of sight (or
collimation). The other components include the tribrach with levelling foot screws, the keyboard with
display and the telescope which is mounted on the standards and which rotates around the tilting axis.
HARISH.T.S, Dept. of Civil Engineering
Page | 9
SURVEYING – II (10CV 44)
Unit – 3 TRIGONOMETRIC LEVELLING
Levelling is carried out in the same way as for a theodolite by adjusting to centralise a plate level or
electronic bubble. The telescope can be transited and used in the face left (or face I) and face right (or
face II) positions. Horizontal rotation of the total station about the vertical axis is controlled by a
horizontal clamp and tangent screw and rotation of the telescope about the tilting axis.
The total station is used to measure angles in the same way as the electronic theodolite.
Distance measurement
All total stations will measure a slope distance which the onboard computer uses, together with the
zenith angle recorded by the line of sight to calculate the horizontal distance.
For distances taken to a prism or reflecting foil, the most accurate is precise measurement. For phase
shift system, a typical specification for this is a measurement time of about 1-2s, an accuracy of (2mm
+ 2ppm) and a range of 3-5 km to a single prism. Although all manufacturers quote ranges of several
kilometres to a single prism.
For those construction projects where long distances are required to be measured, GPS methods are
used in preference to total stations. There is no standard difference at which the change from one to
the other occurs, as this will depend on a number of factors, including the accuracy required and the
site topography.
Rapid measurement reduces the measurement time to a prism to between 0.5 and ’s for both phase
shift and pulsed systems, but the accuracy for both may degrade slightly.
Tracking measurements are taken extensively when setting out or for machine control, since readings
are updated very quickly and vary in response to movements of the prism which is usually pole
mounted. In this mode, the distance measurement is repeated automatically at intervals of less than
0.5s.
For reflector less measurements taken with a phase shift system, the range that can be obtained is
about 100m, with a similar accuracy to that obtained when using a prism or foil.
Keyboard and display
A total station is activated through its control panel, which consists of a keyboard and multiple line
LCD. A number of instruments have two control panels, one on each face, which makes them easier to
use.
In addition to controlling the total station, the keyboard is often used to code data generated by the
instrument – this code will be used to identify the object being measured.
On some total stations it is possible to detach the keyboard and interchange them with other total
stations and with GPS receivers. This is called integrated surveying.
Software applications
The micro processor built into the total station is a small computer and its main function is controlling
the measurement of angles and distances. The LCD screen guides the operator while taking these
measurements.
The built in computer can be used for the operator to carry out calibration checks on the instrument.
The software applications available on many total stations include the following:
-Slope corrections and reduced levels
-Horizontal circle orientation
-Coordinate measurement
-Traverse measurements
-Resection (or free stationing)
-Missing line measurement
-Remote elevation measurement
HARISH.T.S, Dept. of Civil Engineering
Page | 10
SURVEYING – II (10CV 44)
Unit – 3 TRIGONOMETRIC LEVELLING
-Areas
-Setting out
Coordinate measurement
Coordinates of an unknown point relative to a known coordinate can be determined using the total
station as long as a direct line of sight can be established between the two points. Angles and distances
are measured from the total station to points under survey, and the coordinates (X, Y, and Z or
northing, easting and elevation) of surveyed points relative to the total station position are calculated
using trigonometry and triangulation. To determine an absolute location a Total Station requires line
of sight observations and must be set up over a known point or with line of sight to 2 or more points
with known location.
For this reason, some total stations also have a Global Navigation Satellite System interface which do
not require a direct line of sight to determine coordinates. However, GNSS measurements may require
longer occupation periods and offer relatively poor accuracy in the vertical axis.
Angle measurement
Most modern total station instruments measure angles by means of electro-optical scanning of
extremely precise digital bar-codes etched on rotating glass cylinders or discs within the instrument.
The best quality total stations are capable of measuring angles to 0.5 arc-second. Inexpensive
"construction grade" total stations can generally measure angles to 5 or 10 arc-seconds.
Distance measurement
Measurement of distance is accomplished with a modulated microwave or infrared carrier signal,
generated by a small solid-state emitter within the instrument's optical path, and reflected by a prism
reflector or the object under survey. The modulation pattern in the returning signal is read and
interpreted by the computer in the total station. The distance is determined by emitting and receiving
multiple frequencies, and determining the integer number of wavelengths to the target for each
frequency. Most total stations use purpose-built glass corner cube prism reflectors for the EDM signal.
A typical total station can measure distances with an accuracy of about 1.5 millimetres (0.0049 ft) + 2
parts per million over a distance of up to 1,500 metres (4,900 ft).[2]
Reflectorless total stations can measure distances to any object that is reasonably light in color, up to a
few hundred meters.
Data processing
Some models include internal electronic data storage to record distance, horizontal angle, and vertical
angle measured, while other models are equipped to write these measurements to an external data
collector, such as a hand-held computer.
When data is downloaded from a total station onto a computer, application software can be used to
compute results and generate a map of the surveyed area. The new generation of total stations can also
show the map on the touch-screen of the instrument right after measuring the points.
1. ^ Kavanagh, B. F. and Glenn Bird, S. J. (1996). Surveying principles and applications (4 ed.). Prentice
Hall. pp. 257–264. ISBN 0-13-438300-1.
2. ^ Leica Flexline TS02/06/09. (2008). Leica Geosystems. See datasheet for model TS06 (PDF).
Retrieved 27 August 2009
HARISH.T.S, Dept. of Civil Engineering
Page | 11