Surveying I.
Lecture 2.
Sz. Rózsa: Surveying I. – Lecture 1
Outline
Structure of levels
Adjustment of levels
Error sources
Procedure of levelling
Line levelling, detail point levelling
Processing levelling data
Sz. Rózsa: Surveying I. – Lecture 1
The principle of levelling
Line of sight
dA
(lA)
dB
lA
A
lB
(lB)
DHAB
B
DHAB=lA-lB=(lA)-dA-(lB)+dB
When dA=dB (spherical approximation, equal distance to A and B)
DHAB=(lA)-(lB)
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s level
Tilting level
Bubble tube
Diaphragm
Tilting screw
Tilting axis
Circular bubble
Levelling head
Clamping screw - to fix the telescope in one vertical plane
Tangent screw (slow motion screw) - to finely rotate the telescope
along a vertical axis
Sz. Rózsa: Surveying I. – Lecture 1
Elements of Surveyor’s level
How to set the line of sight to be exactly horizontal?
More general: how to set anything to be exactly horizontal?
The bubble tube
Sz. Rózsa: Surveying I. – Lecture 1
The bubble tube
The radius determines the sensitivity of the bubble tube:


R1
R2
R1 greater than R2
Sensitivity: how much the bubble moves due to a given
amount of inclination. The more the bubble moves, the more
sensitive the bubble tube is.
Sz. Rózsa: Surveying I. – Lecture 1
The bubble tube
The determination of sensitivity:


R1
R1
l 2  l1
L

l1
L
l2
  radians
 "   radians

  206264
.8
L
Sz. Rózsa: Surveying I. – Lecture 1
Kepler-type telescope
Object
Eyepiece
Object lens
Virtual image
Note that the virtual image is magnified and inverted!
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s telescope
The diaphragm (cross-hairs)
To provide visible horizontal and vertical reference lines in the
telescope.
Line of collimation
With adjustment screws the diaphragm can be moved in
the telescope to adjust the line of collimation.
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s telescope
Parallax
When focusing the telescope, the real image formed by the objective
lens is made to coincide with the diaphragm.
What is the parallax?
When viewing two distant objects approximately along a straight line, and
the eye is moved to one side, then the more distant object moves relative
to the other in the same direction.
This can lead to observation errors (wrong reading, wrong sighting).
If the real image formed by the objective lens does not coincide with the
diaphragm a parallax is observed -> the reading depend on the position of
the eye!
diaphragm
image
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s telescope
Focusing the telescope
External focusing
Variable length
Focusing lens
Internal focusing
Fixed length
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s level
Tilting level
Bubble tube
Diaphragm
Tilting screw
Circular bubble
Tilting axis
Tribrach
(Levelling head)
Clamping screw - to fix the telescope in one vertical plane
Tangent screw (slow motion screw) - to finely rotate the telescope along a
vertical axis
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s level
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s level
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s level
Tilting level
How can we view the bubble tube?
• Using a mirror (older instrument)
• Prismatic coincidence reader (modern instruments)
Prism
Bubble tube
Bubble tube is tilted
Bubble tube is horizontal (leveled)
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s level
Setting up the level
1. Fix the level on a tripod
2. Center the circular bubble by adjusting the foot screws.
(to approximately level the instrument)
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s level
Setting up the level
3. Sight the levelling staff:
first: rotate the telescope in the direction of the staff
second: use the fine motion screws to ensure precise sighting
(note: on some instruments the fine motion screw works only, when
the alidade is fixed using the fixing clamp)
4. Adjust the levelling bubble using the levelling screw.
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s level
Automatic level
We must adjust the bubble tube before every reading when
using the tilting level -> takes a lot of time, may cause blunders
(large mistakes in the observations)
An automatic level contains an optical device, which
compensates the tilting of the telescope - called compensator.
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s level
Sz. Rózsa: Surveying I. – Lecture 1
The Surveyor’s level
Operation of the compensator
Advantage: faster observations, elimination of a possible
reason of blunders
Disadvantage: vibrations (wind, traffic, etc.) have a bad
impact on the operation of the compensator
Sz. Rózsa: Surveying I. – Lecture 1
The levelling staff
Sz. Rózsa: Surveying I. – Lecture 1
Outline
Structure of levels
Adjustment of levels
Error sources
Procedure of levelling
Line levelling, detail point levelling
Processing levelling data
Sz. Rózsa: Surveying I. – Lecture 1
Adjusting the level
The two-peg test
How much is the collimation error ()?
Collimation error - the line of collimation is not horizontal, when the level is levelled
1. Establish a test line on an approximately flat surface.
2. Compute the elevation difference between the test points (A and B)!
b1
a1
  d1
A
d1
 d2


P
d2
B
The effect of collimation error cancels, when d1=d2.
Thus the height difference is:
DH
AB
 a 1  b1
Sz. Rózsa: Surveying I. – Lecture 1
Adjusting the level
3. Move the instrument to an external point on the extension of the AB line.
4. Compute the elevation difference from the observations (note that the elevation
difference contains the effect of the collimation error)!
b2
a2
 d 1  d 2  d 3 
  d3
A
d1+d2

B
d3
Q
D H AB  a 2   d 1  d 2  d 3   b 2    d 3 
DH
AB
 a 2  b2   d 1  d 2 
5. The true elevation difference is already computed from the previous configuration:
DH
6. Thus the collimation error is:
 
AB
 a 1  b1
a 2  b 2   a1  b1 
d 1  d 2 
Sz. Rózsa: Surveying I. – Lecture 1
Outline
Structure of levels
Adjustment of levels
Error sources
Procedure of levelling
Line levelling, detail point levelling
Processing levelling data
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
The effect of curvature
Line of sight
dA
(lA)
dB
lA
lB
(lB)
DHAB
Solution: Since the equipotential surface is approximately spherical, the effect of curvature
is a function of the instrument-staff distance. When the backsight and foresight distances
are equal, the effect of curvature cancels out.
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
The refraction
The air has different optical properties everywhere. Air pressure, humidity
etc. Have an impact on the refractivity. Thus the light does not propagate
along a straight line, but along a curve:
For points with the same elevation, the effect of refraction can be neglected.
What to do, when they are not?
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
d
r’
radius of refractive curve
dr 
d
dr
2
2r
R  Radius of the Earth
dr 
d
2
R
2r R

d
2
R
2R r
introducin g : k 
R
r
 0 ,13
Solution: the instrument should be set up exactly in the middle
between two points, thus the effect of curvature is the same
for the backsight and foresight.
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
The effect of collimation error
b1
a1


A
d1
P
d2
B
Solution: the instrument should be set up exactly in the
middle between two points and the collimation error must
be constant, thus the effect is eliminated
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
Tilting of the staff

di
The effect depends on the:
• tilting angle
• reading (the higher the reading is, the bigger the error is)
di=li-licos
Solution: staffs should be equipped with circular bubbles
and kept vertical
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
Settlement of the tripod
Measuring the height difference
between A and B!
Measuring the height difference
between B and A!
dh
a1
dh
b1
A
DH
AB
 a 1  b1  d h
b2
a2
B
A
B
D H BA  b 2  a 2  d h
Let’s compute the mean value of the DHAB and DHBA:
D H AB 
D H AB  D H BA
2

a1  b1  d h  b 2  a 2  d h 
2

a1  b1  b 2  a 2 
2

 D H AB   D H BA 
2
Solution: the reading should be taken in both order, and the mean value
of the height differences should be computed (assuming constant
observation speed)
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
Settlement of the staff
Problem:
The staff has a subsidence during the observations. a
change plate must be used to support the staff.
Solution:
- all lines should be run twice in the opposite directions;
- a change plate must be used to support the staff.
Graduation error of the staff
Problem: The cm graduation on the staff is not
accurate. The units have different lengths.
Solution: staffs must be calibrated regularly (the
graduation must be checked in laboratories).
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
Index error of the staff
Problem: The bottom of the staff is not aligned with the 0
unit of the scale.
The effect of the index error on the reading:
l = (l) + d
01
Where l is the reading taken,
while d is the index error
d
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
Staff No. 2.
Staff No. 1.
The effect of index error on a single height difference:
Direction of
levelling
lBS
lFS
DH
DH = lBS-lFS
DH = [(lBS)+d1]-[(lFS)+d2)]=lBS-lFS+d1-d2
When only one staff is used, then the effect of index
error cancels out (d1=d2)
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
What happens when two staffs are used?
Single height difference:
2
Staff No. 1.
1
Staff No. 2.
Staff No. 1.
DH = [(lBS)+d1]-[(lFS)+d2)]=lBS-lFS+d1-d2
The sum of two height differences:
DH = [(lBS)+d1]-[(lFS)+d2)]=lBS-lFS+d1-d2
DH = [(lBS)+d2]-[(lFS)+d1)]=lBS-lFS+d2-d1
Sz. Rózsa: Surveying I. – Lecture 1
Systematic error in levelling
DH1 = [(lBS)+d1]-[(lFS)+d2)]=(lBS)-(lFS)+d1-d2
DH2 = [(lBS)+d2]-[(lFS)+d1)]=(lBS)-(lFS)+d2-d1
DH1 +DH2 = S(lBS)-S(lFS)
When two staffs are used, an even number of stations have to
be created in the levelling line. In this case the effect of the
index error of the staff cancels out.
Sz. Rózsa: Surveying I. – Lecture 1
Outline
Structure of levels
Adjustment of levels
Error sources
Procedure of levelling
Line levelling, detail point levelling
Processing levelling data
Sz. Rózsa: Surveying I. – Lecture 1
Procedure of levelling
1. The instrument must be set up with the same distance to the
staffs.
2. The bubble tube must be levelled before each reading (tilting
level).
3. You must not use the parallax screw between the backsight and
foresight readings
4. The bubble tube must not be affected by strong heat.
5. Readings must be taken 30-50 cm above the ground.
6. Staff should be set up vertically.
7. A change plate should be used to place the staff on the ground.
8. Levelling must be done in two opposite directions.
Sz. Rózsa: Surveying I. – Lecture 1
Procedure of levelling
9. All the observations should be made with a constant speed.
10. Observations should be made only in suitable weather:
cloudy sky, constant temperature, early morning, or late
afternoon.
11. Staff should be calibrated.
12. If there are three hairs in the diaphragm, one should use all
of them to take a reading.
13. When two staffs are used, an even number of stations must
be used to create the levelling line.
Sz. Rózsa: Surveying I. – Lecture 1
Outline
Structure of levels
Adjustment of levels
Error sources
Procedure of levelling
Line levelling, detail point levelling
Processing levelling data
Sz. Rózsa: Surveying I. – Lecture 1
Line levelling
Principle of levelling
Line of sight
dA
(lA)
dB
lA
lB
(lB)
DHAB
What happens, when we want to measure the height difference of two distant
points?
Sz. Rózsa: Surveying I. – Lecture 1
Line levelling
The previous procedure is repeated as many times as need to cover the distance between
the points.
The direction of levelling
Dh1
DH
Dh2
Dh3
Dh4
DH=Dh1+Dh2+Dh3+Dh4
DH=SlBSSlFS
Sz. Rózsa: Surveying I. – Lecture 1
Outline
Structure of levels
Adjustment of levels
Error sources
Procedure of levelling
Line levelling, detail point levelling
Processing levelling data
Sz. Rózsa: Surveying I. – Lecture 1
Processing Levelling Data
Line levelling (one-way)
A
B
HA
HB=?
MSL
Reference level
Sz. Rózsa: Surveying I. – Lecture 1
Line Levelling – one way (the Rise&Fall Method)
d=19
d=20m
A
HA
PID
d=15
d=13
1
2
d
A
BS
FS
12
14
3
Rise
B
Fall
HB=?
H
103.455
1
20
08
33
14
58
0.244
2
3
19
14
74
13
99
0.566
15
08
69
09
13
B
13
11
25
0.561
0.256
0.561
102.950
1.066
DHAB=SRise-SFall=-0.505 m
Sz. Rózsa: Surveying I. – Lecture 1
Line Levelling – two-way (the Rise&Fall Method)
PID
A
1
2
3
B
d
BS
12
08
14
08
20
19
15
13
FS
14
33
74
69
Rise
Fall
H
103.455
14
13
09
11
58
99
13
25
0.244
0.566
0.561
0.256
DHAB=SRise-SFall=-0.505 m
B
1
2
3
A
12
10
13
15
11
13
18
22
03
01
53
22
09
15
09
11
11
19
41
97
0.292
-0.518
0.412
0.325
DHBA=SRise-SFall=+0.511m
Let’s compute the mean height difference:
D H AB 
D H AB  D H BA
2

 0 . 505  0 . 511
2
  0 . 508 m
HB=103.455-0.508=102.947m
Sz. Rózsa: Surveying I. – Lecture 1
Detail Point Levelling – The Height of Collimation Method
Detail Point Levelling: The elevation of some detail points (characteristic
points of objects) should be determined.
A
B
HA
HB
The elevation of the characteristic points
of the ditch should be determined!
MSL
Reference level
Sz. Rózsa: Surveying I. – Lecture 1
Detail Point Levelling – The Height of Collimation Method
Height of collimation: The elevation of the horizontal line of sight. It
can be computed by adding the elevation of the backsight point and the
backsight reading.
Sz. Rózsa: Surveying I. – Lecture 1
Levelling - Bookkeeping
Rise and fall method:
Sz. Rózsa: Surveying I. – Lecture 1
Levelling - Bookkeeping
Height of Collimation method:
Sz. Rózsa: Surveying I. – Lecture 1
Thanks for the Attention!
Sz. Rózsa: Surveying I. – Lecture 1