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TACHEOMETRY

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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

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

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Tacheometry

Concept Determine distances indirectly using triangle

geometry Methods

Stadia Establish constant angle and measure length of

opposite side Length increases with distance from angle vertex

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Stadia System

The theodolite/auto level is directed at the level staff and the distance is measured by reading the top and bottom stadia hairs on the telescope view.

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Stadia System

In the first type the distance between the two. There are two types of instruments used for stadia surveying.

Stadia hairs in the theodolite telescope is fixed.

In the second type of equipment the distance between the stadia hairs is variable, being measured by means of a micrometer.

The most common method used involves the fixed hair tacheometer, or theodolite.

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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

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

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Tacheometry: Stadia

L1

d1

L2

d2

)tan(0.5

0.5Ld 1

1 α

)tan(0.5

0.5Ld 2

2 α

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Stadia Readings

Middle Hair

Upper Hair

Lower Hair

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Stadia Principles

A,B rod intercepts a, b stadia hairs S = rod intercept F = principal focus of

objective lens

C d

D

i

c f

• C = stadia constant• K = f/i = stadia interval factor• d = distance from focal point

to rod• D = distance from instrument

center to rod

b

a a'

b'

F

B

A

S

• f = focal length• i = stadia hair spacing • c = distance from instrument

center to objective lens center

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Stadia Equations

Horizontal sights

100SH

0C100,K usually

CKSH

0V

α100ScosHCcosααKScosH2

2

sin2α100SVCsinαsin2αKSV

21

21

• Inclined sights

i

S

f

d

• From similar triangles

CKSD KSSi

fd

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Tacheometry: Subtense

L1

d1

L2

d2

)tan(0.5

0.5Ld

11 α

)tan(0.5

0.5Ld

22 α

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Subtense Equation Derive equation for computing distance by subtense

L

)2δ0.5Lcot()2δtan(

0.5Ld

• What value would you choose for L?

d

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Centre of instrument

o

Object lensb

x

a

staff

c

f

U V

D

A

X

B

FS i

The notes below shows the calculation of the distance (D) from the centre of the fixed hair tacheometer to a target.

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From the diagram, triangles AOB , a O b are similar

ab

AB

V

U

Ox

OX

Also if OF = f = focal length of object lens

Then + = (lens equation) and multiply both sides by (U f)

U = f + f

U = f + f

1

U

1

V

1

f

UV

AB

ab

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AB is obtained by subtracting the reading given on the staff by the lower stadia hair from the top one and is usually denoted by s (staff intercept), and ab the distance apart of the stadia lines is denoted by i.

This value i is fixed, known and constant for a particular instrument.

U = s + f

D = s + ( f + c )

f

i

f

i

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The reduction of this formula would be simplified considerably if the term (f / i) is made some convenient figure, and if the term (f + c) can be made to vanish.

D = C.S + k

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Constant determination

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

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On a fairly level site chain out a line 100 to 120m long, setting pegs at 25 to 30 meter intervals.

Set at up at one end and determine two distances using tacheometer or theodolite, one short and one long. hence C and K may be determined.

I.E D1 (known) = C.S1 (known) + k D2 (known) = C.S2 (known) + k

Where the instrument designed with an anallatic lens the additive constant k = 0

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For example: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

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.

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ө

ө S

D

V

hi

hL

A

B

Case of inclined sightsVertical elevation angle:

∆L

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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

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hi

D

ө

S

V

h∆L

A

B

Vertical depression angle:

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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

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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 , it is required to:I ) find the horizontal distance to the staff points and their reduced levels.II) find the 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)

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N

D

A

B

C

H1

H2

H3

ө1

ө2

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Solution

For line DA

S1 = 2.20 – 1.10 = 1.10 m

H1 = 100 x 1.10 x Cos2 (+5o 12’) = 109.0964 m

V1 = 109.0964 x tan (+5o 12’) = + 9.929 m

R.L.of A = 150.10 + 1.40 + 9.929 – 1.65 =159.779 m.

For line DB

S2 = 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.

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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(

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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(

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D D

S

S

ӨӨ

Tangential system

Horizontal line of sight :

D = S / tan Ө

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Inclined line of sight :

S

D D

Ө1

Ө2

Ө1

Ө2

D = S / ( tan Ө2 – tan Ө1 )

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Subtense bar system

1 m 1 m

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2 mtheodolite

Subtense bar

α

plan

D = cot( α / 2 )

For distance up to 80 m

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α1 α2

D1 = cot (α1/2) D2 = cot (α2/2)

D = D1 + D2

For distance 80 – 160 m

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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

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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

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Electronic Tacheometry

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.

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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.

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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

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Tacheometry Field Procedure

1. Set up the instrument 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).

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Error Sources

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

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