Tachymetry survey POLITEKNIK MELAKA

C 2005 / 2 / 1
ENGINEERING SURVEY 2
Article I.
Article II.
MODULE
MALAYSIAN
POLYTECHNICS
MINISTRY OF EDUCATION
UNIT 1UNIT 1

ENGINEERING SURVEY 2 C 2005 / 1 /
TACHYMETRY
OBJECTIVES
General Objective : To know and understand the basic concepts of distance
measurement.
Specific Objectives : At the end of the unit you should be able to :-
 Explain the basic concepts of Optical Distance
Measurement.
 Discuss the system that has been use in tachymetry.
 Calculate the distance by using the tachymetry system.
 Explain the procedure to implement the field work
 Explain the steps to process the observation data.
 List errors in tachymetry survey.
2
U
NI

INPUTINPUT
 Explain the application of tachymetry in land surveying
1.1 INTRODUCTION
The word tachymetry is derived from the Greek takhus metron meaning ‘swift
measurement’. It is a branch of surveying where height and distances between ground
marks are obtained by optical means only. An example of tachymetry method is the
stadia method. This method employs rapid optical means of measuring distance using a
telescope with cross hairs (Figure 1.1) and a stadia rod (one stadium = about 607 feet).
The distance between marks can be obtained without using a tape. The tachymeter is any
theodolite adapted, or fitted with an optical device to enable measurement to be made
optically.
Figure 1.1 Two Types of Stadia Hair
1.2 PRINCIPLES OF OPTICAL DISTANCE MEASUREMENT
The tachymetry measurements are based on a common principle. Consider an
isosceles triangle; the perpendicular bisector of the base is directly proportional to the
length of this base. If the base length and paralactic angles are known, then the length of
the perpendicular bisector can be calculated. (Figure 1.2)
3
Cross
Hair
reticle
i =
Stadia
Interval

Figure 1.2 Isosceles Triangle Geometry
(Source : Ukur Kejuruteraan Asas, Abdul Hamid Mohamed)
Distance AB = ½ (Cd) x Cot α/2
If distance AB = D, distance Cd = S , so
Whereby
D = distance between two point
S = base line
α = paralactic angle
1.3 TACHYMETRY SYSTEM
The alternatives of the tachymetry system are classified based on the basic
principles, which are:
a) Fixed angle:
1) The stadia system
i) Incline Sights With The Staff Vertical
ii) Incline Sights With The Staff Normal
4
D = ½ S Cot α/2

b) Variable angle
1) tangential system – vertical staff
2) subtence system – horizontal staff
The theodolite is a standard instrument in each case. It is modified to suit the conditions.
1.3.1 The Stadia System
The diaphragm in this system contains two additional horizontal lines known as
stadia hairs. It is placed equidistant above and below the main horizontal cross hair
(Figure 1.3). The distance between these stadia hair is called the stadia interval (Figure
1.1). This stadia interval is usually a constant, providing fixed-hair tachymetry. This
interval may be altered on some instruments and the movement being measured on a
micrometer.
Figure 1.3 The View In The Telescope
(Source: Ukur Kejuruteraan Asas, Abdul Hamid Mohamed)
Observations are made on to a leveling staff which acts as the variable base. In the
telescope’s field of view the stadia subtend a certain length of the staff or called staff
intercept, which is greater the farther off the staff is held. The staff intercept is
proportional to its distance from the instrument and so from this observed length of the
staff the distance between it and the tachymeter can be obtained.
5

1.3.1.1 The Stadia Formula
The stadia method of providing the horizontal distance between instrument and
staff is shown in Figure 1.4. This technique is always used in stadia tachymetry for
engineering survey. The telescope consists of two centring tubes. The eyepiece and
diaphragm are built at the end of tube. Move the object glass which is built at the other
side when doing focusing.
When the telescope is in focus, the image of the staff AB will be formed at ab in
the plane of the diaphragm. Then a ray of light will emerge parallel to the optical axis
similarly with the ray from B as shown. The rays here will form two similar triangles
each with their apex at F, the base of the smaller triangle at the object glass being equal to
the stadia interval i.
Eyepiece
Diaphragm
Vertical
axis
Picket
Figure 1.4 Stadia Principle
(Source Land Surveying, Ramsay J.P. Wilson)
f --- the focal length of the object glass
F – the outer focal point of the object glass
i --- the stadia interval ab
I--- the distance from the outer focal point to the staff
D---the horizontal distance required
s--- the staff intercept AB
c---the distance from object glass to instrument axis
From these similar triangles:
6

i
f
s
l
=
but l = D – (f + c),
So, the stadia formula:
i
f
s
c)(f-D
=
+
The term f / i is a constant in the stadia formula and is known as the stadia or
multiplying constant and may be denoted by the letter K. The term ( f + c) partly of the
constant f and partly of the variable c, which varies as the object lens is moved in
focusing. However the variation in c is small, especially for sights greater than 10m, and
for all practical purposes may also be considered a constant. The term ( f + c), usually
about 300 to 450mm in this telescope, is known as the additive constant and may be
denoted by the letter C. This reduces the stadia formula to the simple linear equation:
CKsD +=
1.3.1.2 The Analactic Lens
Do you know who J. Porro is?
He is the man who invented the analactic lens in 1840.
In order to save the labour of multiplying the staff intercept each time and the
adding the constant for the particular instrument, it would obviously be simpler if K were
to be 100 and C zero. This would provide a stadia formula of D = 100s and calculation
would merely consist of moving the decimal point of the staff intercept reading two
places to the right. Most of the vernier instruments still in use today do not have an
accurate K value of 100, but most modern tachymeters generally do. In 1840, the
elimination of the additive constant was achieved by an Italian, J. Porro, when he
invented the analactic lens. The inclusion of a second convex lens fixed in relation to the
object glass had the effect of bringing the apex of the measuring triangle, the analactic
7
s
i
f
cfD =+− )(
)( cfs
i
f
D ++=

point, into exact coincidence with the vertical axis of the instrument, as illustrated in
Figure 1.5.
Figure 1.5 The analectic Telescope
(Source : Land Surveying, Ramsay J.P. Wilson)
The term f / i = 1/100 become K = 100. Distance for f and c become similar but in the
opposite side. Therefore C = 0. The stadia formula would now become KsD = , the
additive constants are eliminated. This externally focusing telescope is known as an
analactic telescope.
1.3.1.3 Evaluation of Stadia Constants
In most modern surveying telescopes the stadia constant is designed to be 100 and
the additive constant 0. To confirm the value of these constants or to establish the stadia
of an old or a new instrument, the following fieldwork should be carried out (Figure 1.6)
Figure 1.6 Evaluation of Stadia Constants, K and C
(Source: Asas Ukur Kejuruteraan, Abdul Hamid Mohamed)
8
Object glass
Focus point
Analactic point
Analactic lens
Diaphragm

a) Choose a fairly level ground
b) Set out four pegs A, B, C, and D on that ground. AB is 100m, AC is 40m and AD
is 90m.
c) Set up the tachymeter over the peg at A and observe to a staff that held at C.
d) Not the staff intercepts.
e) Transfer the staff to D and note the staff intercepts.
Distance Stadia Reading Staff Intercept
40
90
1.620, 1.420,1.220
1.871,1.421,0.971
0.400
0.900
Table 1 Obsevation Data
( Source: Asas Ukur Kejuruteraan, Abdul Hamid Mohamed)
The observation data is shown in table 1. K and C can be calculate by using the
stadia formula, D = Ks + C. D is the distance between staff and the tachymeter, s stands
for staff intercept.
40 = 0.4 K + C ------------------------------ (1)
90 = 0.9 K + C ------------------------------ (2)
Now, we can solve the problem by using simultaneous equation.
(2) – (1) 90 – 40 = 0.9 K – 0.4 K
50 = 0.5 K
K
0.5
50
=
K = 100
Replace K =100 in (1)
40 = 0.4 ( 100) + C
C = 40 -40
C = 0
1.3.1.4 Inclined Sight
9

As height differences between staff positions and instrument increase, it will
become impossible to use the horizontal line of sight which so far has only been
considered. In such case a tachymeter must be used to provide an inclined line of sight
and the angle of elevation or depression must be recorded. The stadia formula must now
reflect the angle of inclination of the line of sight and two such cases arise:
a) where the staff is held vertically at the far station
b) where the staff is held to the line of sight from the instrument
1.3.1.4.1 Incline Sights With The Staff Vertical
Figure 1.7 shows that an observation of an inclined sight to a staff held vertically.
A, X and B are the readings on the staff and A’, X and B’ are those which would have
been taken had the staff been swung about X to position it at right-angles or normal to the
line of sight.
In figure 1.7,
s = the staff intercept AB
h = the length of the centre hair reading from the staff base
V = the vertical component XY, the height of the centre hair reading above
(or below) the instrument axis
D = the length of the line of sight IX
H = the horizontal distance required.
H I = instrument height
Figure 1.7 Incline Sight With The Staff Vertical
10

(Source : Land Surveying, Ramsay J.P. Wilson)
From the stadia formula D = Ks + C, it can be seen that the term s in this case is
the distance A’B’ normal to the line of sight. However, the observed value of s is the
length AB, so A’B’ actually equal s, cos θ almost exactly. Therefore the length of the
inclined sight D = Ks + C , but H, the horizontal distance actually required, obviously
equals D= cosθ , therefore the stadia formula now becomes:
H = Ks cos2
θ + C cos θ
From the right angled triangle IXY can been seen that:
V = D sin θ
But D = Ks cos θ + C
V = Ks cos θ sin θ + C sin θ
But cos θ sin θ = ½ sin 2θ
∴V = ½ Ks sin 2θ + C sin θ
In instruments where the additive constant is zero and K = 100, these formulae are
simplified as follows:
H = 100s cos2
θ
V = (100/2) s sin 2θ
To obtain the reduced level at the staff position where the reduced level of the
instrument station is known, the height difference between the points is applied as
follows:
Difference in height, dH = H. I. ± V –h
Where H.I = the height of instrument (always positive)
V = the vertical component (positive for angles of the elevation, negative
for angles depression)
h = the centre hair reading (always negative)
The reduced level of the instrument position I plus the difference in height equal
the reduced level of the staff position S. Therefore:
R.L.s = R.L.I + H.I ± V – h
11

Example 1:
In this example, the value of hi cannot be seen on the rod due to some
obstruction. Here, a rod reading of 2.72 with a vertical angle of -6º 37’ was
booked, along with the h of 1.72 and a rod interval of 0.241., Calculate the
horizontal distance and the vertical distance. Then find the elevation for station 3.
Figure 1
Solution:
H = 100s cos2
θ
= 100 x 0.241 x cos2
6º 37’
= 23.8m
V = (100/2) s sin 2θ
= 100 s cos θ sin θ
= 100 x 0.241 x cos 6º 37’x sin 6º 37’
= -2.76m
R.L.3 = R.L.2 + H.I ± V – h
= 185.16 + 1.72 +- 2.76 – 2.72
= 181.40
12

So, the elevation for station 3 is 181.40,
1.3.1.4.2 Incline Sights With The Staff Normal
Figure 1.8 shows the observation on a staff held normal to an inclined line of
sight. The same notation applies as in figure 1.7.
Figure 1.8 Incline Sight With The Staff Normal to The Line of Sight
(Source: Land Surveying, Ramsay J.P. Wilson)
This time the staff reading normal to the line of sight is the actual reading and does not
have to be reduced as in the previous case. Therefore
D = Ks + C
But H= D kos θ ± (the distance from point X to the vertical through the staff base)
H = (Ks + C) cos θ ± h sin θ
As before V = D sin θ, therefore:
V = (Ks + C) sin θ
In instruments where the additive constant C is zero, K = 100 and the value of θ is less
than 10º (the assumption is generally made that the term h sin θ is zero), these formulae
can be simplified as :
13

H = 100 s cos θ
V = 100 s sin θ
To obtain the reduced level at the staff station, then the height difference between the
points is first reduced as follows:
Difference in height, dH = H. I.± V – h cos θ
1.3.2.3 Comparison of Methods
Conditions Staff Normal/Staff Vertical
a) When
holding staff
♣ Staff can be held vertically with greater ease than in the normal
position.
♣ It’s simpler to plumb a staff with a staff bubble than hold the staff
normal to a line of sight.
♣ For normal holding, it needs to be attached with a peep-sight
perpendicular to the face of the staff, so the staff-man can sight
towards the instrument.
♣ In bush the peep-sight may be obscured, preventing normal holding,
while the upper part of the staff is still available for sighting in the
vertically held position.
♣ The normal position may also be found by swinging the staff until the
lowest possible reading of the centre cross hair is obtained. However,
it is difficult to signal to the staff-man the correct position in the bush.
Conditions Staff Normal/Staff Vertical
b) Reduction
of
observation
♣ The vertical staff reduction formulae are simpler than the normal
staff reduction formula when the h sin θ and h cos θ are included
in the normal formulae.
c) Careless
staff holding
♣ Errors of distance and elevation are very much more marked when
there is a deviation from the normal position especially on steep
sights.
The normal position may also be found by swinging the staff until the lowest
possible reading of the centre cross hair is obtained. However, it is difficult to signal to
the staff-man the correct position in the bush.
14

1.3.2 The Tangential System
In this system the paralactic angle subtended by a known length of staff is
measured directly. Figure 1.9 shows the method where observations are taken to an
ordinary levelling staff held vertically.
Figure 1.9 The Tangential System of Tachymetry
(Source: Land Surveying, Ramsay J.P. Wilson)
The instrument is set up and the vertical circle is read on both faces to give the
angle of elevation (or depression) to a whole staff graduation. This process is repeated to
another whole graduation to give as large a staff intercept, s, as possible. From the staff
intercept and the two observed vertical angles θ and φ, the horizontal distance H may be
calculated as:
AY = H tan θ
BY = H tan φ
AY –BY = s
= H ( tan θ – tan φ)
∴
)tan(tan φθ −
=
s
H
or
)tan(tan θφ −
=
s
H ( for sight down hills)
The difference in height between the instrument station and the staff station is found as
follows :
15

Vertical component, V = BY = H tan φ
Height difference, dH = H.I ± V – BX
Check : AY=H tan θ ( when dH = H.I. ± V – AX)
1.3.3 The Substance Bar
The substance system is a particular form of the tangential system where the
measured base is held horizontally as illustrated in Figure 1.10 instead of vertically. The
paralactic angle is measured with greater accuracy using the horizontal circle instead of
the vertical circle The horizontally held base is especially made for this purpose and is
known as a substance bar.
The substance bar is a specially made instrument supported on a tripod with two
sighting targets set a precise distance apart, usually 2m. The central target in the
substance bar is placed midway between the end targets for traverse angle measurement
and for use in sighting with the auxiliary base method. The sighting device is fixed at
right-angles to the line of the bar so that it may be positioned at right angle to the line of
sight from the theodolite.
As temperature affects the bar length, the subtence bar targets are usually attached
to invar rods or wires, which have a low coefficient of expansion, so that their nominal
distance apart remains almost constant. The targets may be lit from behind for night
observations, which have the advantage of a less disturbed atmosphere resulting in
increased accuracy in the angular measurement.
16

Figure 1.10 The Principles of Horizontal Distance Measurement
( Source : Ukur Kejuruteraan Asas, Abdul Hamid Mohamed)
From the figure above, it can be seen that the horizontal distance
2
cot
2
αs
H =
because half the bar length divided by the perpendicular sector of the isosceles triangle of
half the measured angle α. Usually the bar is 2m long to simplify the calculation. So
2
cot
α
=H . As the paralactic angle is measured on the horizontal plane, the distance
obtained is always the horizontal distance and no slope corrections are ever necessary
however far above or below the theodolite the substance bar may be.
If height differences between theodolite and bar stations are required then a
vertical angle θ must be measured to the line of the bar and the vertical component
calculated from the formula V = H tan θ (figure 1.11). The height of the theodolite above
its station (Hi) and the height of the bar above its station (Hb) must be measured. Then
the height difference between stations X and Y(dHXY) is shown as below:
dH = Hi ±V – Hb
where Hi = Height of the theodolite
V = vertical component
Hb = Height of substance bar
So, the reduced level of the staff position Y, RL x = RL Y + dHXY
17

Figure 1.11 The Height Difference Between Stations
( Source : Ukur Kejuruteraan Asas, Abdul Hamid Mohamed)
18

Activity 1a
1.1 What is meant by the term ‘tachymetry’?
1.2 Explain the basic principles upon which tachymetric measurement are based?
1.3 A vertical staff is observed with a horizontal external focusing telescope at a
distance of 112.489m. Measurements of the telescope are recorded as :
Objective to diaphragm 230mm
Objective to vertical axis 150mm
If the readings taken to the staff were 1.073, 1.629 and 2.185, calculate
a) the distance apart of the stadia lines (i)
b) the multiplying constant (K)
c) the additive constant (C)
1.4 What are the main differences between the stadia system and tangential system?
19

Feedback 1a
1.1 Tachymetry means swift measurement where height and distances between
ground marks are obtained by optical means only.
1.2 The tachymetry measurements are based on the common principle of the isosceles
triangle. The perpendicular bisector of the base is directly proportional to the
length of this base. If the base length and paralactic angle are known, then the
length of the perpendicular bisector can be calculated.
20
Try your best to
answer these
questions.

Distance AB = ½ (Cd) x Cot α/2
If distance AB = D, distance Cd = S ,
so
D = ½ S Cot α/2
Whereby
D = distance between two point
S = base line
α = paralactic angle
1.3 From equation D = Ks + C
= )( cfs
i
f
++
)( cfD
fs
i
+−
=
)150.0230.0(489.112
)073.1185.2(230
+−
−
=
380.048.112
)100.1(230
−
=
= 2.3 mm
Therefore, 100
3.2
230
===
i
f
K
21

INPUTINPUT
C = f +c
= 230+150
= 380mm
1.4
In the stadia system, the apex angle of the measuring triangle is defined by the
stadia hairs on the telescope diaphragm. The base length is obtained by observing
the intersection of the stadia hairs on the image of the measuring staff seen in the
telescope’s field or view. The tangential system in which the apex angle
subtended by a basic of known length is accurately measured, usually with the
single-second theodolite. In order to obtain the distance between instrument and
base, the tangent of the angle or angles observed must be used in the calculation.
Well done. You have done a good job!!
1.4 STADIA FIELD PRACTICE
Stadia tachymetry is mainly used in surveying details in selected areas. Adequate
horizontal and vertical control, supplied by traversing and leveling is required to orientate
the survey and to provide station levels. It is best suited to open ground where few hard
levels are required.
22

In field practice, the transit is set on a point for which the horizontal location and
elevation have been determined. If necessary, the elevation of the transit station can be
determined after setup by sighting on a point of known elevation and working backward
through equation elevation station (Rod) = elevation station (instrument) + Hi - h ± V. Hi
is instrument height, V is the vertical component and h is the centre hair reading
1.4.1 PROCEDURE OF FIELD WORK
Figure 1.12 shows an area which needs topographic survey. There are some object
illustrated in that figure, such as station (A), building(B), road(C), fence(D) and
drainage(E). The procedures below show the way to implement the stadia field works.
a) Establish 4 control stations (station 1, station 2, station 3 and station 4) by
using wooden pegs.
b) Implement horizontal control networks on each station in order to obtain the
coordinate for every station. Record the data in a field book.
c) After that, implement the leveling process to get the elevation of each station.
Enter the observations in the field book.
d) Now, use either stadia tachymetry method or stadia electronic method to set
the theodolite over station 2.
e) Measure the height of the theodolite at station 2 as Hi2 with a steel tape.
f) Set the horizontal circle to zero.
g) Sight the reference station (Station 1) at 0º00’.
h) Sight the stadia point to Station 3 by loosening the clamp (clamp is tight).
i) Sight the main horizontal hair roughly on the value of h, then move the lower
hair to the closest even foot (decimeter) mark.
j) Read the upper hair, determine the rod interval, and enter the value in the
notes.
k) Sight the main horizontal hair precisely on the h value.
l) Wave off the rod holder on point a, point b and point c.
m) Read and book the horizontal angle and the vertical angle from station 2 to
points a, b and c. Try to take as many details as possible.
n) Check the zero setting for the horizontal angle before moving the instrument
to station 3.
o) Repeat step d to m for observation at station 3(3-d, 3-e,3-f) , station 4 (4-g,4-
h,4-i) and station1(1-j).
23

p) Finally reduce the notes (compute horizontal distances and elevation) after
field hours and check the reductions.
Figure 1.12 Stadia Field Works.
(Source: Ukur Kejuruteraan 1 , Baharin Mohammad)
24

C 2005 / 2 /
ENGINEERING SURVEY 2
1.4.2 Recording of Observation
Stadia tachymetry is best booked in tabulated form as below.
Station and
Instrument
Height
Horizontal
Angle (α)
Vertical
angle
Middle
Stadia
reading
Stadia
reading
(a –upper
reading)
b- lower
reading)
Horizontal
Length
H = Ks
Cos2
θ-C
Vertical
difference
V=(Ks Sin
2θ)/2 - sinθ
Difference
in Height
ΔH = Hi
±V-h
Reduced
level of
station
Reduced
level of
point
Remarks
Station 2
1.542m
50 23 00 +88 31 0.6m a- 0.890
b- 0.310
57.961m 1.501m 2.443m 100 102.443m Station 1-
control station
343 25 00 -92 32 0.5m a- 0.551
b- 0.449
10.068m -1.153m -0.111m 99.889m a- beside
drainage
342 57 00 - 96 36 0.4m a-0.454
b-0.346
10.657m -1.233m -0.091m 99.909m b- beside
drainage
357 00 00 -96 20 0.8m a-0.837
b- 0.763
7.340m -0.811m -0.069m 99.931m c- road side
305 31 00 -94 28 1.2m a-1.242
b-1.548
8.353m -0.652m -0.310m 99.690m d-tree (radius-
2.7m)
214 16 00 -94 37 1.2m a- 1.230
b- 1.170
5.961m -0.481m -0.139m 99.861m e- beside
drainage
220 37 00 -94 05 1.3m a- 1.326
b- 1.274
5.174m -0.369m -0.127m 99.873m f- beside
drainage
250 36 00 -94 06 1.3m a- 1.334
b- 1.266
6.765m -0.485m -0.243m 99.757m g- lamp post
255 26 00 -94 23 1.3 a- 1.323
b- 1.277
4.573m --0.351m -0.109m 99.891m h- road side
Table1.1 Tachymetry Stadia Method Booking
25

Explanation of the booking
Column 1 : Station number and height of instrument
Column 2 : The bearing of the ray oriented on the control points
Column 3: Vertical angle (θ) or the zenith angle. Z = (θ = 90-Z)
For example point 1 Z = 93°, then θ = 90°- 93°= 3° 00'
Column 4: Middle stadia reading.
Column 5 : Upper and lower stadia readings
Column 6 : H, the horizontal length = KsCos2θ - C ( Usually 100sCos2θ) by using the
data from column 3 and 5.
Column 7 : Vertical difference = H tan θ or
(usually 50 sin 2θ - C) by using the data from column 3 and 5.
Column 8: Difference in height, which is calculated from formula Hi ±V-h
Column 9 : Axis level of the station
Column 10 : Reduced level of the point : axis level ± (V-h)
Column 11 : Remarks amplification of diagram.
1.5 ACCURACY OF STADIA OBSERVATION
a) Accuracy of distance measurement
Under ideal conditions, it should be possible to obtain an accuracy of 0.01% in distance
measurement, but this is seldom achieved in practice. Using an ordinary levelling staff
with 10 mm divisions practical accuracies approximate to the following:
Distance 20m 100m 150m
Accuracy ±100mm ±200mm ±300mm
b) Accuracy of height measurement
Provided that the staff is held vertically with reasonable care and angles of sighting are
less than 10˚, then heights should be accurate to within 0.01 per cent of the sighting
distance.
1.5.1 Errors in horizontal distances
• The error of careless staff holding can be resolved by using staff bubbles
when implementing field observation.
• Error in reading the stadia intercept, which is immediately multiplied by
100(K1), thereby making it significant. This source of error will increase
C
Ks
−
2
2sin θ

with the length of sight. The obvious solution is to limit the length of sight
to ensure a good resolution of the graduations.
• Error in the determination of the instrument constants K1 and K2, resulting
in an error in distance directly proportional to the error in the constant K1
and directly as the error in K2.
• Effect of differential refraction on the stadia intercepts. This is minimized
by keeping the lower reading 1 to 1.5m above the ground.
• Random error in the measurement of the vertical angle. This has a
negligible effect on the staff intercept and consequently on the horizontal
distance.
In addition to the above sources of error, there are many others resulting from
instrumental errors, failure to eliminate parallax, and natural errors due to high winds and
summer heat. The lack of statistical evidence makes it rather difficult to quote standards
of accuracy; however, the usual treatment for small errors will give some basis for
assessment.
1.5.2 Errors in elevations
The main sources of error in elevation are errors in vertical angles and additional
errors rising from errors in the computed distance. Figure 1.13 clearly shows that whilst
the error resulting from errors in vertical angles remains fairly constant, the results from
additional errors rising from errors in the computed distance increases with increased
elevation.
θtanDH =
( ) ( )[ ]
( ) ( )[ ]
046.0
"1sin"205sec2005tan48.0
sectan
sec
tan
222
2/1222
2
±=
×°+°±=
+±=∴
=
=∴
θδθθδδ
θδθδ
θδδ
DDH
DH
DH
This result indicates that elevation need be quoted only to the nearest 10mm.
Accuracies of 1 in 1000 may still be achieved in tachymetry traversing, due to the
compensating effect of accidental errors, reciprocal observation of the lines and a general
increase in care.
27

Figure 1.13 Errors In Elevation.
(Source : Engineering Surveying, W. Schofield)
1.6 PLOTTING
After the field observations, the collected data must now be processed. The
traverse closure is calculated and then all adjusted values for northing, easting and
elevations are computed manually or by computer programs. After that, the data is
processed by using software such as TRPS and Autocad. All the details can be plotted the
following way.
a) Plotting Control Station.(Figure 1.14)
♪ Place the control station on a grid paper.
♪ The grid paper is printed with grid lines at 1mm intervals.
♪ When plotting on grid paper, the stations are defined by using the
coordinate system.
♪ Station 1 is assumed as the origin whereby coordinate x is 1000m and
coordinate y 1000m. The position of station is plotted starting from the
lower left corner of the grid paper.
♪ Scaling along the x-axis from coordinate x station 1, plot the coordinate of
station 2, X2. Using the same way also plot coordinate Y2. Repeat this
step for station 3 and station 4
28

Figure 1.14 Determination Of Station Position On Grid Paper.
b) Detailing (Figure 1.15)
♪ Instead of using coordinates, plotting can be done by scaling the bearing and
distances of a detail.
♪ Normally a protractor and a scale ruler are needed in plotting.
♪ Place the circular protractor with its centre station 2 and the zero lined up
with the reference station 2-1.
♪ Mark the bearing of θa on the paper against the protractor edge.
♪ Remove the protractor and draw the direction of the line 2-a. Scale the
distance and plot the position of a.
♪ Repeat the same steps when marking off point c-j.
♪ Finally, join the points to form the detail.
29

Figure 1.15 Details Plotting On Grid Paper.
c) Contours
♪ After marking all the details, the contours need to be plotted.
♪ Contours are lines on a map representing a line joining points of equal
height on the ground. This method is most commonly adopted for larger
areas.
♪ Reduced level is placed beside details and the height of each point is
spotted.
♪ Finally, contour lines are plotted by using the interpolation method.
d) Preparation of Title Block on Tracing Paper.
♪ All topography and engineering drawings have title blocks.
♪ Usually the title block is placed in the right corner of the plan, which also
has the logo client, project name, date of project and other details.(Figure
1.16).
30

♪ Revisions to the plan are usually referenced immediately above the title
block, showing the date and a brief description of the revision.
♪ The title block is often of standard size and has a format similar to that in
figure 2.4.
♪ All the drawings on tracing paper are done manually by using the
technical pens with Indian ink or by using AutoCad software.
♪ The size of the technical pen is determined based on texts and lines
required in a drawing.
Figure 1.16 Title Block
e) Final drawing.
♪ After completing the title block, transfer all the drawings from the grid
paper to tracing paper.
♪ Then, write the additional text or draw lines and symbols in that drawing.
♪ The texts refer to the name of building, road, and values of height and
contour intervals.
♪ Plot the text horizontally for all values except the value of height.
♪ Every detail has lines of different types and sizes. (For example, the root
line of hedge is shown in black and the outline in green)
♪ Finally, plot the details by using a plotter. (Figure 1.17).
31
Scale
Direction
Grid value
Logo client
Plan number
Plan title
Datum explanation
Legend
Explanation of observation, Land Survey
Firm, Name of surveyor, date, plan reference
number and others.

Figure 1.17 Details That Plot On A 8-Pen Plotter.
(Source: Surveying With Construction Application, B.F. Kavanagh)
1.7 APPLICATION
This method is easy to apply in the field, but unless a direct-reading tachymeter is
used, the resultant computation for many ‘spot-shots’ can be extremely tedious, even with
the use of a computer program. The very low order of accuracy and its short range limit
its application to detail surveys in rural areas or contouring.
1.7.1 Detail Survey
The theodolite is set up at a control station A ( Figure 1.18) and oriented to any
other control station (RO) with the horizontal circle set to 0º 00’. Thereafter the bearings
(relative to A–RO) and horizontal length to each point of detail (P1, P2, P3, etc) are
obtained by observing the stadia readings on a staff held there, the horizontal circle
reading (φ1, φ2,φ3, etc) and the vertical angle. The cross hair-reading is also required
to compute the reduced level of the point.
The field data is booked as shown in table 2.1. Note that the angles are required to the
nearest minute or arc only. It is worth noting that the staff-man should be the most
experienced member of the survey party who would appreciate the error sources, the limit
accuracy available and thus the best and most economic staff positions required.
32

At station A
Grid ref E 400, N300
Weather Cloudy,cool
Stn level (RL) 30.84m OD
Ht of inst (hi) 1.42m
Axis level (RL + hi) 31.90m(Ax)
Survey Canbury Park
Surveyor J. SMITH
Date 12.12.83
Staff
point
Angles observed Staff
readings
Staff
intercept
Horizontal
Distance
Ks cos2
θ
Vertical
Angle
K/2 s sin
2 θ
Reduced
level
Remarks
Hori-
zontal
Vertical
circle
Vertical
angle
° ′ ° ′ ° ′ s D ± V Ax ± V-h
RO 0 00 Station B
P1 48º12’ 95º20’ -5º20’
1.942
1.404
0.866
1.076 106.67 -9.96 20.54 Edge of
pond
P2 80º02’ 93º40’ -3º40’
0.998
0.640
0.281
0.717 71.41 -4.58 26.68 Edge of
pond
P3 107º56’ 83º20’ +6º40’
1.610
1.216
0.822
0.788 77.74 +9.09 39.77 Edge of
pond
Table 1.2 Booking Of Field Data
Figure 1.18 Detail Survey
33

1.7.2 Contouring
Contouring is carried out exactly the same manner as above, but with many more
spot shots along each radial arm (Figure 1.19). The arms are turned off at regular angular
intervals, with the staff –man obtaining levels at regular paced intervals along each arm
and at each distinct change in gradient. Subsequent computation of the field data will fix
the position and level of each point along each arm, which may then be interpolated for
contours.
Figure 1.19 Contouring
1.8 FURTHER OPTICAL DISTANCE-MEASURING EQUIPMENT
1.8.1 Direct-Reading Tachymeters
Direct-reading tachymeters or self reducing tachymeters as they are also called,
have curved lines replacing the conventional stadia lines. Figure 1.20 illustrates one
particular make, in which the outer lines are curves to the function cos2
θ and the inner
curves are to the function sinθ cosθ. Thus the outer curve staff intercept is not just S but
S cos2
θ. Hence, one need only multiply and intercept reading by K1=100 to obtain the
horizontal distance. Similiarly, the inner curve staff intercept is S sinθ cosθ, and need
only be multiplied by K1 to produce the vertical height H. The separation of the curves
varies with variation in the vertical angle.
34

There are other makes of instruments which have different methods of deriving at
the solution. However, the objective remains the same-to eliminate computation. It
should be noted that there is no improvement in accuracy.
Figure 1.19 Outer Lines of Direct-reading tachymeters
(Source: Engineering Surveying, W. Schofield)
1.8.2 Vertical-Staff Precision Tachymeter.
The vertical-staff tachymeter as produced by Kern and named the Kern DK-RV,
has a moveable diaphragm which varies with the inclination of the telescope, the amount
of variation being controlled by a gear-and-cam mechanism. It is used with a specially-
graduated vertical staff giving horizontal distances to an accuracy of 1 in 5000 over a
maximum range of 150m. Figure 1.20 illustrates a portion of the special staff as viewed
through the instrument. By rotating the telescope in the vertical plane, the horizontal
reticule A is made to bisect the zero wedge. Rotation of the instrument in azimuth is
carried out until the sloping reticule B bisects a small circular dot on the left-hand scale.
The instrument now reads as follows:
Reticule B = 15.00m
Vertical reticule C = 0.88m
Horizontal distance = 15.88m
The same comments apply to this instrument as to the horizontal-staff precision
tachymeter.
35

Figure 1.20 Portion of Vertical-Staff Precision Tachymeter.
(Source: Engineering Surveying, W. Schofield)
1.8.3 Total Station
When electronic theodolites are combined with interfaced EDMIs and an
electronic data collector, they become electronic tachymeter instruments- Total Stations.
The total stations can read and record horizontal and vertical angles together with the
slope distances. The microprocessors in the total stations can perform a variety of
mathematical operations, for example, averaging multiple angle measurement, averaging
multiple distances measurement, determining X, Y, Z coordinates and others. The data
collected can be handled by a device connected by cable to the tachymeter but many
instruments come with the data collector built into the instrument.
Data are stored on board internal memory about (1300-points) and on memory
cards (about 2000 points per card). The data can be directly transferred to the computer
from the total station via cable, or the data transferred from the data storage cards first to
a card reader-writer and from there to the computer. This section will be discussed further
in Unit 6.
36

Figure 1.21 Total Station
(Source: Surveying With Construction Application, B.F. Kavanagh)
37

Activity 1b
1.5 List down 5 steps that are needed to produce a topographic map.
1.6 The following observations were taken with a tachymeter, having constants of
100 and zero, from point A to B and C. The distance BC was measured as 157m.
Assuming the ground to be a plane within the triangle ABC, calculate the horizontal
distance and vertical distance for AB.
At To Staff readings (m) Vertical Angle
A B 1.48, 2.73, 3.98 + 7° 36’
C 2.08, 2.82, 3.56 -5° 24’
1.7 In tachymetry survey, the accuracy of stadia observation is affected by several
sources of error. Describe 3 of these errors.
1.8 Describe the procedure to implement the stadia field work in tachymetry survey.
38
Look for the
solutions
now.

Feedback 1b
1.5 There are 5 steps to produce a topographic map.
a) Plotting Control Station
b) Details plotting
c) Contours
d) Preparation of title block on tracing paper.
e) Final drawing
1.6 Horizontal distance AB = 100 x S cos2
θ
= 100 x (3.98 – 1.48) cos2
7° 36’
= 246 m
Vertical distance AB = 246 tan 7° 36’
= +32.8m
1.7 There are three kind of errors:
a) Careless staff holding. This is minimized by using staff bubbles.
b) Error in reading the stadia intercepts. This source of error will increase with
the length of sight. The obvious solution is to limit the length of sight to
ensure good resolution of the graduations.
c) Effect of differential refraction on the stadia intercepts. This is minimized by
keeping the lower reading 1 to 1.5m above the ground.
1.8 Establish 4 control stations (station 1, station 2, station 3 and station 4) by using
wooden pegs.
• Implement horizontal control networks on each. Record the data in a field book.
• After that, implement the levelling process to get the elevation of each station.
Record the observations in the field book.
• Now, use either stadia tachymetry method or stadia electronic method to set the
theodolite over station 2.
• Measure the height of the theodolite at station 2 as Hi2 with a steel tape.
• Set the horizontal circle to zero.
• Sight the reference station (Station 1) at 0º00’.
• Sight the stadia point to Station 3 by loosening the clamp (clamp is tight).
• Sight the main horizontal hair roughly on the value of h, then move the lower hair
to the closest even foot (decimeter) mark.
• Read the upper hair, determine the rod interval, and enter the value in the notes.
• Sight the main horizontal hair precisely on the h value.
39

• Wave off the rod holder on point a, point b and point c.
• Read and book the horizontal angle and the vertical angle from station 2 to points
a, b and c. Try to take as many details as possible.
• Check the zero setting for the horizontal angle before moving the instrument to
station 3.
• Repeat step d to m for observation at station 3(3-d, 3-e,3-f) , station 4 (4-g,4-h,4-i)
and station1(1-j).
• Finally reduce the notes (compute horizontal distances and elevation) after field
hours and check the reductions.
Figure 2
40
I got
it !!!

Self Assessment
1) Tachymetry is used to determine the elevation of the instrument station B base on
elevation of station A. Explain the tachymetry stadia formula below by using
illustrations.
R.L.B = R.L.A + H.I + V – h
Where:
R.L.B = elevation of the instrument station B
R.L.A = elevation of the instrument station A
H.I.= instrument height
h = the length of the centre hair reading from the stsff base
the instrument axis
2) A line of third order levelling is run by theodolite, using tachymetry methods
with a staff held vertically. The usual three staff readings of centre and both stadia
hairs are recorded together with the vertical angle (VA). A second value of height
difference is found by altering the telescope elevation and recording the new
readings by the vertical circle and centre hair only.
The two values of the height differences are then meaned. Compute the difference
in height between the points A and B from the following data:
The stadia constant are :multiplying constant =100; additive constant = 0.
Backsights
VA
Staff Foresights
VA
Staff Remarks (all
measurements in m)
+ 0º 02’ 00” 1.890
1.417 Point A
0.945
+0º 02’ 00” 1.908
-0º 18’ 00” 3.109 Point B
2.012
0.914
0º 00’ 00” 3.161
(height difference between the two ends of theodolite ray = 100s cos θ sin θ, where s=
stadia intercept and θ = VA)
3) The rod reading made to coincide with the value of the hi, is typical of 90 percent
of all stadia measurements. In figure 1, the vertical angle is + 1º 36’ and the rod
41

interval is 0.401. Both the rod hi and the rod reading (R.R.) are 1.72m. Calculate
the horizontal distance and the vertical distance. Then find the elevation for
station 2.
. Figure 2
4) A theodolite has a tachymetry constant of 100 and an additive constant of zero.
The centre of reading on a vertical staff held on a point B was 2.292m when
sighted from A. If the vertical angle was +25° and the horizontal distance AB is
42

Feedback to Self Assessment
190.326m, calculate the other staff readings and thus show that the two intercept
intervals are not equal. Using these values calculate the level of B if A was
37.95m and the height of the instrument 1.35m.
5) The table below shows a tachymetry stadia field booking. Complete the table
below and calculate elevation of each point.
Station
and
instru-
ment
height
Hori-
zontal
Angle
(α)
Vertical
angle
Middle
Stadia
reading
a –upper
reading
b- lower
reading
Horizont
al Length
H = 100s
Cos2
θ-C
Vertical
difference
V=100s
cosθ sinθ
Reduced
level
of station
Reduced
level
of point
Remarks
Station
4
10.417
1.417 45° 51' -90° 18’ 3.100
3.301
2.900
Beside
road
170°18’ -98° 48’ 1.120
1.252
1.000
Lamp post
120°21’ 87° 46’ 2.202
2.475
2.100
Centre
line
43
How to
answer
this???
“Try your best
to find the
solution.”

1)
The figure above shows
R.L.B = R.L.A + H.I + V – h
Where:
R.L.B = elevation of the instrument station B
R.L.A = elevation of the instrument station A.
H.I.= instrument height
h = the length of the centre hair reading from the staff base
the instrument axis
2) V = 100s sin θ cos θ
= 50s sin 2 θ
To A, V = 50 (1.890 -0.945) sin 0º 04’ 00”
= 0.055m
Difference in level from instrument axis = 0.550 – 1.417
= -1.362
Check Reading
V = 50 (0.945) sin 0º 40’ 00”
= 0.550m
Difference in level from instrument axis = 0.550 -1.980
44
Theodolite
Staff

= -1.358
Mean = 1.360m
To B, V = 50 (3.109-0.914) sin -0º 36’ 00”
= -1.149m
Difference in level from instrument axis = -1.149-2.012
= -3.161
Check level = -3.161
Mean = - 3.161m
Difference in level AB = -3.161+1.360
= -1.801m
3) H= 100s cos2
θ
= 100 x 0.401 x cos2
1º 36’
= 40.1m
V = (100/2) s sin 2θ
= 100 s cos θ sin θ
= 100 x 0.401 x cos 1º 36’x sin 1º 36’
= +1.12m
R.L.2 = R.L.1 + H.I ± V – h
= 185.16 + 1.72 +1.12 – 1.72
= 186.28
So, the elevation for station 2 is 186.28.
4)
45

Figure 1
From basic equation, CD = 100s cos2
θ
190.326m = 100 s cos2
25°
s = 2.316m
From figure 1, HJ = s cos2
25°
= 2.316 * cos2
25°
= 2.1m
Inclined distance CE = CD sec cos2
25°
"23'340
210
1.2
2 °==∴ radα
"11'170°=∴ α
Now by reference to figure 1:
DG = CD tan (25°- α )
= 190.326 tan (25° -0° 17’ 11”)
= 87.594
DE = CD tan 25°
= 190.326 tan 25°
= 88.749
DF = CD tan (25° + α )
= 190.326 (25° + 0° 17’ 11”)
= 89.910
It can be seen that the stadia intervals are:
GE = DE – DG
= S1
= 88.749 – 87.954
46

= 1.115
EF = DF – DE
= S2
= 89.910 – 88.749
= 1.161
From which it is obvious that the a) Upper reading = (2.292 +1.161) = 3.453
b) Lower raeding = (2.292 – 1.155) = 1.137
Vertical Height DE = h
= CD tan 25°
= 190.326 tan 25°
= 88.749( as above)
∴ Level of B = 37.95 + 1.35 + 88.749 2.292
= 125.757 m
5)
47
=2.316
(Check)
CD = 100s cos2
θ…..(Bla
bla bla)

Station
and
instru-
ment
height
Hori-
zontal
Angle
(α)
Vertical
angle
Middle
Stadia
reading
a –upper
reading
b- lower
reading
Horizont
al Length
H = 100s
Cos2
θ-C
Vertical
difference
V=100s
cosθ sinθ
Reduced
level
of
station
Reduced
level
of point
Remarks
Station
4
10.417
1.417 45° 51' -90° 18’ 3.100
3.301
2.900
40.099 0.210 8.524
a- Beside
road
170°18’ -98° 48’ 1.120
1.252
1.000
24.610 3.810 6.904
b -Lamp
post
120°21’ 87° 46’ 2.202
2.475
2.100
37.443 1.460 11.092 c-Center line
Reduced level of point a = 10.417 +1.417 -0.210 – 3.100
= 8.524
Reduced level of point b = 10.417 +1.417 -3.810 – 1.120
= 6.904
Reduced level of point c = 10.417 +1.417 + 1.460 – 2.202
= 11.092
48
Congratulations, you can
proceed to the next unit.
IN
P
U
T

Tachymetry survey POLITEKNIK MELAKA

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