Surveying I.

Lecture 4.

Setting up a theodolite

1. Setting up the tripod (the head of the tripod should be approximately horizontal) above the control point

2. Fix the instrument on the tripod.

3. Sight the control point in the optical plummet using the footscrews of the instrument.

4. Level the instrument by adjusting the length of the legs of the tripod using the circular bubble.

5. Find the normal point of the bubble tube.

6. Level the instrument accurately using the bubble tube and the three levelling screws.

7. Finally loosen the instrument on the tripod, and slide it above the control point on the head of the tripod.

8. Fix the instrument on the tripod.

Setting up a theodolite

Systematic errors of angle measurements

Systematic errors can be caused by:

• structural failures or misalignments of the instrument

• the observer

• external conditions during the observation (weather, etc)

Systematic errors can be eliminated by:

• eliminating the reason of the systematic error (adjustment of the instrument, following the rules - weather)

• using suitable measuring procedures

• computing the effect of the error, and correcting the observations

Systematic errors of angle measurements

Let’s suppose that the systematic errors are independent -> can be treated

separately.

Diaphragm is tilted

We have to use the same point to sight a target.

Intersection of the crosshairs.

Systematic errors of angle measurements

Collimation error

The line of sight is not perpendicular to the transit axis.

Systematic errors of angle measurements

The effect of collimation error on the horizontal readings:

Systematic errors of angle measurements

since:

tantan

''

OMcOP

MMPP and:

OP

OMc

tan

tan

When the angles are small:

cc

tan

tanand: cos

OP

OM

cos

c

tan'

tan'

OMMM

cOPPP

Thus:

Systematic errors of angle measurements

Readings in two faces!

Systematic errors of angle measurements

The transit axis is not adjusted

The transit axis is not perpendicular to the standing axis.

1

1sinPO

PP

From the OP’P1’ triangle:

11

1tanPP

PP

From the P’P1P1’ triangle: 1

11tansinPO

PP

since

1

11tanPO

PP

then

tanReadings in two faces!

Systematic errors of angle measurements

Excentricity of the telescope

The line of sight do not intersect the standing axis.

P

V=O

( )

( )

Excen

tric lin

e of s

ight

Correct lin

e of sight

d

arcsin

OPd

Readings in two faces!

Systematic errors of angle measurements

Excentricity of the horizontal circle

O

V

r

”

( )”

( )’( )”

( )’’

i’(exc)

i’(c)i”(c)

i”(exc)

The center of the horizontal circle do not coincide with the standing axis.

Sine theorem:

sinsin

r

sinarcsinr

Dangerous error source: r=14cm, ()=10-6m, =90 -> ()=14,4”

Readings in two facesor

using two indices!

Systematic errors of angle measurements

Tilting of the horizontal circle

The plane of the horizontal circle is not perpendicular to the standing axis.

Graduation error of the horizontal circleCould be neglected for modern instruments. However repeated measurements should be taken by rotating the horizontal circle with 180°/# of repetitions.

Systematic errors of AM - Setting up errors

Centering error

The extension of the standing axis does not go through the station.

V=O

d

ic

iexc

0

0

iexc

sinarcsinr

Centering should be checked

before each repetition!

Systematic errors of AM - Ext. conditions

Deformation of the tripod

The tripod may revolve due to direct and uneven sunlight.

Experience show that the speed of revolution is constant. 0tt

Effect of refraction

The light does not propagate along a straight line.

Majority of the impact is on the vertical angle measurements.

Will be discussed later.

FL and FR readingsin the opposite order!

The Horizontal Reading

Readings are taken in two faces: Face left (FL) <-> Face right (FR)

How can the horizontal reading computed from the FL and FR readings?

0

Face Left

0

Face Right

In theory the difference between FL and FR reading should be exactly 180°.

The Horizontal Reading

BUT: Systematic errors exist - like the collimation error

The effect has opposite sign in the FL and FR reading.

The Horizontal Reading

Example

0

Face Left

0

Face Right

transit axis

P

P’’

P’

M

M’

M’’

cc

FL: 88-05-26FR: 268-05-55

The difference should be 180°, but it is 180°00’29’’.

2 = 29’’ -> = 14,5’’

Horizontal Reading = FL + = 88° 05’ 40’’

The Zenith Angle

Face Left Face Right

0 0

In theory the sum of the FL and FR readings should be 360°.

The Zenith Angle

What happens, when the index is not in the vertical direction?

Face Left

0Index error

Face Right

0Index error

The Zenith Angle

How can we compute the zenith angle?

Face Left

0Index error

Face Right

0Index error

FL: 68-15-22FR: 291-44-20

2=360°-FL-FR=18’’

=9’’

Z = FL+ =68-15-31

Thank You for Your Attention!