ERT247/4 GEOMATICS ENGINEERING TACHEOMETRY

ERT 247 GEOMATICS ENGINEERING 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 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Figure 1 shows the set-up of a tachymetric measurement. ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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. ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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. ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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) ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Tacheometry: Stadia L2 d2  L1 d1 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Stadia Readings Upper Hair Lower Hair Middle Hair ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Stadia Principles C d c f A b b' i S a a' F B D A,B rod intercepts a, b stadia hairs S = rod intercept F = principal focus of objective lens f = focal length i = stadia hair spacing c = distance from instrument center to objective lens center C = stadia constant K = f/i = stadia interval factor d = distance from focal point to rod D = distance from instrument center to rod ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Stadia Equations From similar triangles Horizontal sights Inclined sights ERT 247 GEOMATICS ENGINEERING

Tacheometry: Subtense L 2 d2 L 1 d1 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Subtense Equation Derive equation for computing distance by subtense  L d What value would you choose for L? ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING The notes below shows the calculation of the distance (D) from the centre of the fixed hair tacheometer to a target. staff A Centre of instrument b Object lens S F i x X o c f a U V B D ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING From the diagram, triangles AOB , a O b are similar Also if OF = f = focal length of object lens Then + = (lens equation) and multiply both sides by (U f) U = f + f 1 1 1 U V f U V AB ERT 247 GEOMATICS ENGINEERING ab

ERT 247 GEOMATICS ENGINEERING 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 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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 ERT 247 GEOMATICS ENGINEERING

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 ERT 247 GEOMATICS ENGINEERING

anallatic lens the additive constant k = 0 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 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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 + 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. ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Case of inclined sights Vertical elevation angle: ө S h L V B ө ∆L hi A D ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Vertical depression angle: ө hi V A S ∆L h B D ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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. 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 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING H1 D ө1 H3 ө2 B H2 C ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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. ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING From Triangle DCB BC= BC= 48.133 m From Triangle DAB AB= AB= 83.471 m ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Tangential system Horizontal line of sight : S Ө Ө S D D D = S / tan Ө ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Inclined line of sight : Ө1 Ө1 Ө2 Ө2 S D D D = S / ( tan Ө2 – tan Ө1 ) ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Subtense bar system 1 m 1 m ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING For distance up to 80 m Subtense bar theodolite α 2 m D = cot( α / 2 ) plan ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING For distance 80 – 160 m α1 α2 D1 = cot (α1/2) D2 = cot (α2/2) D = D1 + D2 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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 ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING For distance 350 – 800 m x 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) ] ERT 247 GEOMATICS ENGINEERING

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. ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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. ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING 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 ERT 247 GEOMATICS ENGINEERING

Tacheometry Field Procedure Set up the instrument 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). ERT 247 GEOMATICS ENGINEERING

ERT 247 GEOMATICS ENGINEERING Error Sources There are 4 main sources of error: Staff Readings Tilt of the Staff Vertical Angle Horizontal Angle ERT 247 GEOMATICS ENGINEERING

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