GEOMATICS ENGINEERING
TACHEOMETRY
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
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
L1
L2
d1
d1 d2
d2
0.5L1 tan(0.5α ) 0.5L 2 tan(0.5α )
Stadia Readings
Upper Hair
Middle Hair
Lower Hair
A,B rod intercepts a, b stadia hairs S = rod intercept F = principal focus of objective lens
Stadia Principles
K
c b a
i
d
f
A
b' a'
S 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 d S f i
d
f S KS i
D CS K
• Horizontal sights H CS K
V 0
usually C 100,K 0
H 100S
• Inclined sights H CScos2α Kcosα
H 100Scos2α
V CS 12 sin2α Ksinα
V 100S 12 sin2α
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
0.15
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.
Case of inclined sights Vertical
ө
elevation angle:
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
D
B
D = CS cos2 Ө + K cos Ө ; = 1/2 C S sin 2 Ө + 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: Staff points
bearing
Vertical 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)
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.
N
A H1 D
ө1
H3
ө2
B H2
C Staff points
bearing
Vertical 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 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 =
(109.096) 2 (136.186) 2 2 109.096 136.186 cos190
AC = 48.505 m
From Triangle DCB BC= (130.000)2 (136.186)2 2 1030.000 136.186 cos 200 40 BC= 48.133 m From Triangle DAB AB= (130.000)
2
AB= 83.471 m
(109.096)
2
0
2 1030.000 109.096 cos19
Tangential system
Horizontal line of sight : S Ө Ө
S
D
D = S / tan Ө
D
Inclined Ө1
line of sight :
Ө1 Ө2
Ө2
S
D
D = S / ( tan Ө2 – tan Ө1 )
D
Subtense bar system 1m
1m
For distance up to 80 m Subtense bar theodolite
α
2m
D = cot( α / 2 )
plan
For distance 80 – 160 m α1
α2
D1 = cot (α1/2)
D2 = cot (α2/2)
D = D1 + D2
For distance 160 – 350 m Auxiliary α
Theodolite 1
β
base
900
Theodolite 2 x/2
β α
X = ( 2D )1/2 ; X = cot ( α/2 ) , D = X cot β , D = X/2 cot β/2
x/2
x
For distance 350 – 800 m α β1 β1
β2
β2
D1
X X D D
x
D2
= 0.7( 2D )1/2 ; = cot ( α/2 ) , = X ( cot β1 + cot β2 ) , = X/2 [ cot (β1/2) + cot (β2/2) ]
x/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.
reflector
D Hr A
Ө
HI B
S
S = D cos Ө R.L.of point A = R.L.of point B + H I + V - Hr
V
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
