Surveying I. Lecture 2. Sz. Rózsa: Surveying I. – Lecture 1

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



The principle of levelling

A

B

(lA)

(lB)

A

HAB

lA

B

lB

HAB=lA-lB=(lA)-A-(lB)+B

When A=B (spherical approximation, equal distance to A and B)

HAB=(lA)-(lB)

topography

equipotentialsurface

Line of sight

The Surveyor’s level

Tilting level

Levelling head

Tilting screw

Diaphragm

Bubble tube

Tilting axis

Clamping screw - to fix the telescope in one vertical plane

Tangent screw (slow motion screw) - to finely rotate the telescope along a vertical axis

Circular bubble


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


The bubble tube

The radius determines the sensitivity of the bubble tube:

R2R1

R greater thanR1 2

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.


The bubble tube

The determination of sensitivity:

R1

L

l1

R1

L

l2

radians

L

ll 12

8.206264" radians


Kepler-type telescope

Object lens

Eyepiece

Object

Virtual image

Note that the virtual image is magnified and inverted!


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.



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



Focusing the telescope

External focusing

Internal focusing

Focusing lens

Variable length

Fixed length



Tilting level

Tribrach (Levelling head)

Tilting screw

Diaphragm

Bubble tube

Tilting axis

Clamping screw - to fix the telescope in one vertical plane

Tangent screw (slow motion screw) - to finely rotate the telescope along a vertical axis

Circular bubble







Tilting level

How can we view the bubble tube?

• Using a mirror (older instrument)• Prismatic coincidence reader (modern instruments)

Bubble tube

Prism

Bubble tube is tilted Bubble tube is horizontal (leveled)

Bubble tube



Setting up the level1. Fix the level on a tripod

2. Center the circular bubble by adjusting the foot screws.(to approximately level the instrument)



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.



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.





Operation of the compensator

Advantage: faster observations, elimination of a possible reason of blundersDisadvantage: vibrations (wind, traffic, etc.) have a bad impact on the operation of the compensator


The levelling staff


Outline

Structure of levels


Error sources





Adjusting the level

The two-peg test

d1 d2

a1b1

A BP

1d 2d

Collimation error - the line of collimation is not horizontal, when the level is levelled

The effect of collimation error cancels, when d1=d2.

Thus the height difference is: 11 baH AB


How much is the collimation error ()?

1. Establish a test line on an approximately flat surface.

2. Compute the elevation difference between the test points (A and B)!

Adjusting the level

321 ddd

3d

323212 dbdddaH AB

d +d1 2

a2b2

A B d3 Q

2122 ddbaH AB

11 baH AB 5. The true elevation difference is already computed from the previous configuration:

21

1122

dd

baba


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)!

6. Thus the collimation error is:

Outline

Structure of levels


Error sources





Systematic error in levelling

The effect of curvature

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.


(lA)

(lB)

A

HAB

lA

B

lB

topography


Line of sight


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?



13,0:

22

2

22

2

r

Rkgintroducin

r

R

R

d

R

R

r

d

EarththeofRadiusRr

d

r

r

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.


d

r’radius of refractive curve

r


The effect of collimation error

d1 d2

a1b1

A BP

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



Tilting of the staff

The effect depends on the:• tilting angle• reading (the higher the reading is, the bigger the error is)

Solution: staffs should be equipped with circular bubbles and kept vertical


i

i=li-licos


Settlement of the tripod

hbaH AB 11 habHBA 22

Solution: the reading should be taken in both order, and the mean value of the height differences should be computed (assuming constant observation speed)


A B

h

a1 b1

Measuring the height difference between A and B!

Measuring the height difference between B and A!

A B

h

a2 b2

Let’s compute the mean value of the HAB and HBA:

2222

22112211 BAABBAABAB

HHabbahabhbaHHH


Settlement of 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

Solution: staffs must be calibrated regularly (the graduation must be checked in laboratories).


Problem: The staff has a subsidence during the observations. a change plate must be used to support the staff.

Problem: The cm graduation on the staff is not accurate. The units have different lengths.


Index error of the staff

Problem: The bottom of the staff is not aligned with the 0 unit of the scale.

01

The effect of the index error on the reading:

l = (l) +

Where l is the reading taken, while is the index error



The effect of index error on a single height difference:

H = lBS-lFS

H = [(lBS)+1]-[(lFS)+2)]=lBS-lFS+1-2

When only one staff is used, then the effect of index error cancels out (1=2)


Direction oflevelling

lBS

lFS

Sta

ff N

o.

1.

Sta

ff N

o.

2.

H


What happens when two staffs are used?

Single height difference:

The sum of two height differences:

1

2


H = [(lBS)+1]-[(lFS)+2)]=lBS-lFS+1-2

Sta

ff N

o.

1.

Sta

ff N

o.

2.

Sta

ff N

o.

1.

H = [(lBS)+1]-[(lFS)+2)]=lBS-lFS+1-2

H = [(lBS)+2]-[(lFS)+1)]=lBS-lFS+2-1


H1 +H2 = (lBS)-(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.


H1 = [(lBS)+1]-[(lFS)+2)]=(lBS)-(lFS)+1-2

H2 = [(lBS)+2]-[(lFS)+1)]=(lBS)-(lFS)+2-1

Outline

Structure of levels


Error sources






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.



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.


Outline

Structure of levels


Error sources





Line levelling

Principle of levelling

What happens, when we want to measure the height difference of two distant points?


(lA)

(lB)

A

HAB

lA

B

lB

topography


Line of sight

Line levelling

The previous procedure is repeated as many times as need to cover the distance between the points.

H=h1+h2+h3+h4

The direction of levelling

H

h1

h2

h3

h4


H=lBSlFS

Outline

Structure of levels


Error sources





Processing Levelling Data


Line levelling (one-way)

A

B

MSLReference level

HA HB=?

A

BHA HB=?


PID d BS FS Rise Fall HA

1

1

d=20m

20

12 14

14 58 0.244

103.455

2

2

d=19

19

08 33

13 99 0.566

d=15

3

3 15

14 74

09 13 0.561

d=13

B 13

08 69

11 25 0.256

0.561 1.066

HAB=Rise-Fall=-0.505 m

102.950

Line Levelling – one way (the Rise&Fall Method)

PID d BS FS Rise Fall HA 12 14 103.4551 20 08 33 14 58 0.2442 19 14 74 13 99 0.5663 15 08 69 09 13 0.56

1B 13 11 25 0.256

B 12 031 11 10 01 09 11 0.29

22 13 13 53 15 19 -0.5183 18 15 22 09 41 0.41

2A 22 11 97 0.32

5


Line Levelling – two-way (the Rise&Fall Method)

HAB=Rise-Fall=-0.505 m

HBA=Rise-Fall=+0.511m

Let’s compute the mean height difference:

mHH

H BAABAB 508.0

2

511.0505.0

2 HB=103.455-0.508=102.947m


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

MSLReference level

HA HB

The elevation of the characteristic points of the ditch should be determined!


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.

Levelling - Bookkeeping

Rise and fall method:


Levelling - Bookkeeping

Height of Collimation method:


Thanks for the Attention!


Documents

Surveying I. Lecture 2. Sz. Rózsa: Surveying I. – Lecture 1