LOGO ERT247/4 GEOMATICS ENGINEERING TACHEOMETRY MS SITI KAMARIAH MD SA ’ AT LECTURER SCHOOL OF BIOPROCESS ENGINEERING [email protected].

LOGOERT247/4

GEOMATICS ENGINEERING

TACHEOMETRY

MS SITI KAMARIAH MD SA’ATLECTURER

SCHOOL OF BIOPROCESS [email protected]

What is tacheometry??

Easy and cheap method of collecting much topographic data.

Tachymetry (or tacheometry) also called “stadia surveying” in countries like England and the United States

means “fast measurement”; rapid and efficient way of indirectly measuring distances and elevation differences

Figure 1 shows the set-up of a tachymetric measurement.

Tacheometry

Concept Determine distances indirectly using triangle

geometryMethods

Stadia• Establish constant angle and measure length of

opposite side• Length increases with distance from angle

vertex

Stadia System

The theodolite/auto level is directed at the level staff

the distance is measured by reading the top and bottom stadia hairs on the telescope view.

Measurement

Electronic Tacheometry: Uses a total station which contains an

EDM, able to read distance by reflecting off a prism.

Subtense Bar system: An accurate theodolite, reading to 1" of

arc, is directed at a staff, two pointings being made and the small subtended angle measured

Equipment

Measurement can be taken with theodolites, transits and levels and stadia rods

While in the past, distances were measured by the “surveyor’s chain” or tape

This can be done easier and faster using a telescope equipped with stadia hairlines in combination with a stadia rod (auto level and staff)

Tacheometry: Stadia

L1

d1

L2

d2

)tan(0.5

0.5Ld 1

1 α

)tan(0.5

0.5Ld 2

2 α

Stadia Readings

Middle Hair

Upper Hair

Lower Hair

Stadia Principles

A,B rod interceptsa, b stadia hairsS = rod intercept F = principal focus of objective lens

K d

D

i

c f

K = stadia constantC = f/i = stadia interval factord = distance from focal point to rodD = distance from instrument center

to rod

b

a a'

b'

F

B

A

S

f = focal lengthi = stadia hair spacing c = distance from instrument

center to objective lens center

Stadia Equations

• Horizontal sights

100SH

0K100,Cusually

K CSH

0V

α100ScosH

KcosααCScosH2

2

sin2α100SV

Ksinαsin2αCSV

21

21

• Inclined sights

i

S

f

d

• From similar triangles

KCSD KSSi

fd

In practice, the multiplicative constant generally equals 100 and the additive constant equals zero.

This is certainly the case with modern instruments by may not always be so with older Theodolites.

The values are usually given by the makers but this is not always the case.

It is sometimes necessary to measure them in an old or unfamiliar instrument.

The simplest way, both for external and internal focusing instruments, is to regard the basic formula as being a linear one of the form:

D = C.S + K

Constant determination

For example:

D =C.S + K30.00 = 0.300 * C + K90.00 = 0.900 * C + Ktherefore C = 100 & K = 0

Any combination of equations gives the same result, showing that the telescope is anallatic over this range, to all intents and purposes.

Distance Readings Intervals

(m) upper Stadia

Centre Lower Stadia

upper lower total

30.000 1.433 1.283 1.133 0.15 0.15 0.30

55.000 1.710 1.435 1.160 0.275 0.275 0.55

90.000 2.352 1.902 1.452 0.450 0.450 0.90

ө

ө

S

D

V

hi

hL

A

B

Case of inclined sights

Vertical elevation angle:

∆L

L = C S cos Ө + K ,

D = L cos Ө

Then ;

D = CS cos2 Ө + K cos Ө ;

V = L sin Ө = ( C S cos Ө + K ) sin Ө

= 1/2 C S sin 2Ө + K sin Ө ;

∆L = h i + V – h = R.L. of B - R.L. of A ;

Where : h is the mid hair reading

hi

D

ө

S

V

h∆L

A

B

Vertical depression angle:

D = CS cos2 Ө + K cos Ө ;

= 1/2 C S sin 2Ө + K sin Ө ;

∆L = - h i + V + h = R.L. of A - R.L. of B ;

Where : h is the mid hair reading ;

Ө may be elevation or depression

Example

From point D three points A, B and C have been observed as follows:

If the reduced level of D is 150.10 m. , hi = 1.40 m. and the tacheometeric constant = 100 , find:i) the horizontal distance to the staff points and their reduced levels.ii) distance AB , BC , and CA.

Staff points

bearingVertical angles

Stadia readings

A 85º 30΄ 5º 12΄ (1.10,1.65,2.20)

B 125º 10΄ 0 (2.30,2.95,3.60)

C 104º 30΄ 9º 30΄ (1.45,2.15,2.85)

N

D

A

B

C

H1

H2

H3

ө1

ө2

Staff points

bearingVertical angles

Stadia readings

A 85º 30΄ 5º 12΄ (1.10,1.65,2.20)

B 125º 10΄ 0 (2.30,2.95,3.60)

C 104º 30΄ 9º 30΄ (1.45,2.15,2.85)

Solution

For line DAS1 = 2.20 – 1.10 = 1.10 mH1 = 100 x 1.10 x Cos2 (+5o 12’) = 109.0964 mV1 = 109.0964 x tan (+5o 12’) = + 9.929 mR.L.of A = 150.10 + 1.40 + 9.929 – 1.65 =159.779 m.For line DBS2 = 3.60 – 2.30 = 1.30 m.H2 = 100 x 1.30 x Cos2 (+00.00) = 130 m.V2 = 130 x tan (+00.00) = + 00.00 m.R.L. of B =150.10 + 1.40 + 00.00 – 2.95 = 148.55 m.

For line DCS3 = 2.85 – 1.45 = 1.40 m.H3 = 100 x 1.40 x Cos2 (+9o 30’) = 136.186 m.V3 = 136.186 tan (+9o 30’) = + 22.790 m.R.L. of C = 150.10 + 1.40 + 22.79 – 2.15 = 172.140 m.θ1 = 104o 30’ – 85o 30’ = 19o 00’θ2 = 125o 10’ – 104o 30’ = 20o 40’θ = 19o 00’ + 20o 40’ = 39o 40’

From Triangle DAC

AC = AC = 48.505 m

022 19cos186.136096.1092)186.136()096.109(

From Triangle DCB BC=

BC= 48.133 m

From Triangle DAB

AB=AB= 83.471 m

0420cos186.136000.10302)186.136()000.130( 022

022 19cos096.109000.10302)096.109()000.130(

D D

S

S

ӨӨ

Tangential system

Horizontal line of sight :

D = S / tan Ө

Inclined line of sight :

S

D D

Ө1

Ө2

Ө1

Ө2

D = S / ( tan Ө2 – tan Ө1 )

Subtense bar system

1 m 1 m

2 mtheodolite

Subtense bar

α

plan

D = cot( α / 2 )

For distance up to 80 m

α1 α2

D1 = cot (α1/2) D2 = cot (α2/2)

D = D1 + D2

For distance 80 – 160 m

Theodolite 1Theodolite 2

900β

αAuxiliary

base

x/2

x/2β

α

x

X = ( 2D )1/2 ;

X = cot ( α/2 ) , D = X cot β , D = X/2 cot β/2

For distance 160 – 350 m

For distance 350 – 800 m

D1 D2

X = 0.7( 2D )1/2 ;X = cot ( α/2 ) , D = X ( cot β1 + cot β2 ) ,D = X/2 [ cot (β1/2) + cot (β2/2) ]

β1β2

α x

x/2β1 β2

Electronic Tacheometry (Total Station)

The stadia procedure is used less and less often these days, more commonly geomatic engineers use a combination theodolite-EDM known in jargon as a total station.

Often these instruments are connected to a field computer which stores readings and facilitates the processing of the data electronically.

Electronic Tacheometry

This instrumentation has facilitated the development of this method of detail and contour surveying into a very slick operation.

It is now possible to produce plans of large areas that previously would have taken weeks, in a matter of days.

The math's behind the operation is very simple, it is in effect the same as the stadia formulae with the term for the distance replaced by the measured slope distance.

reflector

Ө

HI

D

S

HrV

S = D cos Ө

R.L.of point A = R.L.of point B + HI + V - Hr

B

A

Tacheometry Field Procedure

1. Set up the instrument (Theodolite) at a reference point

2. Read upper, middle, and lower hairs.3. Release the rodperson for movement to

the next point.4. Read and record the horizontal angle

(azimuth).5. Read and record the vertical angle

(zenith).

Error Sources

There are 4 main sources of error: Staff Readings Tilt of the Staff Vertical Angle Horizontal Angle