ERT247/4 GEOMATICS ENGINEERING TACHEOMETRY MS SITI KAMARIAH MD SA’AT LECTURER SCHOOL OF BIOPROCESS ENGINEERING sitikamariah@unimap.edu.my

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 geometry Methods 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 L2 d2  L1 d1

Stadia Readings Upper Hair Lower Hair Middle Hair

Stadia Principles A,B rod intercepts a, b stadia hairs S = rod intercept F = principal focus of objective lens Stadia Principles K d c f A b b' i S a a' F B D f = focal length i = stadia hair spacing c = distance from instrument center to objective lens center K = stadia constant C = f/i = stadia interval factor d = distance from focal point to rod D = distance from instrument center to rod

Stadia Equations From similar triangles Horizontal sights Inclined sights

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

For example: Distance Readings Intervals (m) upper Stadia Centre Lower Stadia upper lower total 30.000 1.433 1.283 1.133 0.15 0.30 55.000 1.710 1.435 1.160 0.275 0.55 90.000 2.352 1.902 1.452 0.450 0.90 D =C.S + K 30.00 = 0.300 * C + K 90.00 = 0.900 * C + K therefore 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.

Case of inclined sights Vertical elevation angle: ө S h L V B ө ∆L hi A D

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

Vertical depression angle: ө hi V A S ∆L h B D

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. Stadia readings Vertical angles bearing Staff points (1.10,1.65,2.20) 5º 12΄ 85º 30΄ A (2.30,2.95,3.60) 125º 10΄ B (1.45,2.15,2.85) 9º 30΄ 104º 30΄ C

N A H1 D ө1 H3 ө2 B H2 C Stadia readings Vertical angles bearing Staff points (1.10,1.65,2.20) 5º 12΄ 85º 30΄ A (2.30,2.95,3.60) 125º 10΄ B (1.45,2.15,2.85) 9º 30΄ 104º 30΄ C

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.

For line DC S3 = 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

From Triangle DCB BC= BC= 48.133 m From Triangle DAB AB= AB= 83.471 m

Horizontal line of sight : Tangential system Horizontal line of sight : S Ө Ө S D D D = S / tan Ө

Inclined line of sight : Ө1 Ө1 Ө2 Ө2 S D D D = S / ( tan Ө2 – tan Ө1 )

Subtense bar system 1 m 1 m

plan For distance up to 80 m D = cot( α / 2 ) Subtense bar theodolite

For distance 80 – 160 m D = D1 + D2 α1 α2 D1 = cot (α1/2)

X = cot ( α/2 ) , D = X cot β , D = X/2 cot β/2 For distance 160 – 350 m Auxiliary base x α β 900 Theodolite 1 Theodolite 2 x/2 β α x/2 X = ( 2D )1/2 ; X = cot ( α/2 ) , D = X cot β , D = X/2 cot β/2

For distance 350 – 800 m X = 0.7( 2D )1/2 ; α β2 β1 x/2 β1 β2 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) ]

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.

R.L.of point A = R.L.of point B + HI + V - Hr reflector D Hr V A Ө HI B S S = D cos Ө R.L.of point A = R.L.of point B + HI + V - Hr

Tacheometry Field Procedure Set up the instrument (Theodolite) at a reference point Read upper, middle, and lower hairs. Release the rodperson for movement to the next point. Read and record the horizontal angle (azimuth). 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

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