Surveying I.

Lecture 2.

Outline

Levelling

Structure of levels

Adjustment of levels

Error sources

Procedure of levelling

Levelling

MSL - Mean Sea LevelLevel line - a line with constant height (a curved line)Horizontal line - a tangential to the level line at the instrument

MSL

lineLevelh

linehorizontal

curvatureofeffect

ncollimatioofLine

Levelling

Over short distances the horizontal line and level line coincide.For a distance of 100m the effect of the curvature is less than 1 mm.

The levelling device (called level) must be set up so, that the line of sight is perpendicular to the gravity vector (plumb line). -> the line of sight is horizontal.

sightofline

Horizontal

heightin

Difference

staffGraduated

staffGraduated Level

Levelling

Levelling

Levelling - Bookkeeping

Rise and fall method:

Levelling - Bookkeeping

Height of Collimation method:

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

The bubble tube

The determination of sensitivity:

R1

L

l1

R1

L

l2

radians

L

ll 12

8.206264" radians

The Surveyor’s 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.

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 targeting).

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

The Surveyor’s telescope

Focusing the telescope

External focusing

Internal focusing

Focusing lens

Variable length

Fixed length

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

The Surveyor’s level

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)

The Surveyor’s level

Setting up the level

Primary axis

Secon

dary

axis

1. Fix the level on a tripod

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

3. Sight the levelling staff, and eliminate the parallax.

4. Adjust the sensitive bubble tube by the tilting screw.

The Surveyor’s level

Automatic levelWe must adjust the bubble tube before every reading when using the tilting level -> takes a lot of time, may cause blunders

An automatic level contains an optical device, which compensates the tilting of the telescope - called compensator.

The Surveyor’s level

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

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, if d1=d2.

Thus the height difference is:

11 baH AB

Adjusting the level

The two-peg test

321 ddd

3d

323212 dbdddaH AB

d +d1 2

a2b2

A B d3 Q

2122 ddbaH AB

11 baH AB From the previous configuration:

21

1122

dd

baba

Systematic errors in levelling

MSL

lineLevelh

linehorizontal

curvatureofeffect

ncollimatioofLine

The effect of curvature

Solution: the instrument should be set up exactly in the middle between two points.

Systematic errors 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:

Systematic errors in levelling

r

d

13,0:

22

2

22

2

r

Rkgintroducin

r

R

R

d

R

R

r

d

EarththeofRadiusRr

d

r

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.

Systematic errors in levelling

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

Systematic errors in levelling

Tilting of the staff

l

lDepends on the:• tilting angle• reading (the higher the reading is, the bigger the error is)

Solution: staffs should be equipped with circular bubbles

Systematic errors in levelling

Settlement of the tripod

a1b1

A B

h

hbaH AB 11

a2b2

A B

hbaH AB 22

Solution: the reading should be taken in both order, and the mean value of the height differences should be computed

Orders:• backsight - foresight• foresight - backsight

Systematic errors in levelling

Settlement of the staff

Solution: all lines should be run twice in the opposite directions.

Procedure of levelling

1. The instrument must be set up in the middle between two staffs.

2. The bubble tube must be levelled before each reading.

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.

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.

Thanks for the Attention!