tacheometric surveying. unit i tacheometric surveying contents tacheometric surveying tangential,...

TACHEOMETRIC SURVEYING

2

UNIT I Tacheometric Surveying

CONTENTS

• Tacheometric Surveying• Tangential, Stadia and sub-tense methods• Stadia Systems• Horizontal and inclined sights• Vertical and Normal Staffing• Fixed and movable hairs• Stadia constants• Anallactic lens• Subtense bar

Sivapriya Vijayasimhan

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Introduction

• Tacheometry – Greek word means quick measure• Measuring horizontal and vertical distance of a points on the earth

surface relatively to one another are determined without using a chain or tape or a separate levelling instrument.

• Preparation of contoured maps or plans with higher accuracy and also, it provides a check on distances measured with the tape.

• Need of Tacheometry : steep and broken ground, deep revines, stretches of water or swamp etc., where chaining is difficult or impossible


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Uses of Tacheometry

Measuring horizontal distances and differences in elevations.

1. Preparation of topographic maps which require both elevations and horizontal distances

2. Survey work in difficult terrain where direct methods are inconvenient

3. Detail filling

4. Reconnaissance surveys for highways, railways, etc.

5. Checking of already measured distances

6. Hydrographic surveys

7. Establishing secondary controlSivapriya Vijayasimhan

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Instrument- Transit theodolite fitted with a stadia diaphragm

In addition to it, convex lens (anallatic lens)is provided between the eye-piece and the object glass at a fixed distance.

Various pattern of stadia diaphragm

-Levelling Staff-Stadia Rod


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Instrument

- Transit theodolite fitted with a stadia diaphragm

-The stadia diaphragm essentially consists of one stadia hair above and the other an equal distance below the horizontal cross-hair, the stadia hairs being mounted in the ring and on the same vertical plane as the horizontal and vertical cross-hairs.

(1) The simple external-focusing telescope

(2) The external-focusing anallactic telescope (Possor`s telescope)

(3) The internal-focusing telescope.


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Tangential System

1. Diaphragm of the tacheometer is not provided with stadia hair

2. Single Horizontal Hair is used

3. Staff consist of two vanes at known distances

4. Two points are required to measure staff intercept

5. Angles, elevations or depressions are measured

6. Tangents are used to measure horizontal distances and elevations

(not generally used)


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Stadia SystemPrinciple: Tacheometric angle is constant

Staff intercept varies with distance between staff and instruments, which forms base

1. Diaphragm of tacheometer is provided with two stadia hair (upper and lower)2. Telescope is directed towards the staff held at a point whose distance from instruments is

to be found3. Difference in these readings gives staff intercept4. Horizontal distance is obtained by multiplying staff intercept by multiplying constant

Two Methods1.Fixed hair 2.Movable Hair Method


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Subtense Method1. Reverse of stadia method2. Staff intercept forms fixed base3. Tacheometric angle according with staff position

Fixed Base : fixed distance between two tangents or vanesInterval between the stadia wires is changed till lines of sight coincide with tangents and the subtended angle is noted

Base may be vertical or Horizontal

Base Vertical : Movable Hair method or Vertical base subtense method

Vertical angle is measured with the help of special diaphragm – high accuracy

Base Horizontal : Horizontal base subtense method Horizontal angle is measured by method of repetition using Transient theodolite

example: Subtense bar method


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Principle of TacheometerBased on Isoscales Triangle

Ratio of

k = ½ cot β/2

f – focal lengthi- stadia interceptSivapriya Vijayasimhan

A

B

QPO

S3S2S1

R

D1D2

D3

A’

B’O’

i β


Sivapriya Vijayasimhan 11

Tacheometer must essentially incorporate the following features:(i) The multiplying constant should have a nominal value of 100 and the error contained in

this value should not exceed 1 in 1000. (ii) The axial horizontal line should be exactly midway between the other two lines. (iii) The telescope should be truly anallactic. (iv) The telescope should be powerful having a magnification of 20 to 30 diameters.

• The aperture of the objective should be 35 to 45 mm in diameter to have a sufficiently bright image.

• For small distances (say upto 100 meters), ordinary levelling staff may be used. For greater distances a stadia rod may be used.

• A stadia rod is usually of one piece, having 3 – 5 meters length. • A stadia rod graduated in 5 mm (i.e. 0.005 m) for smaller distances and while for longer

distances, the rod may be graduated in 1 cm (i.e. 0.01 m).


10' 7"

Common Patterns of Stadia Rods

LC of the stadia rods are less than the LC of ordinary Levelling Staff

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Stadia Tacheometry


A

Object glass O

B

CB’C’A’

}}F Vi

f

d

v u

A, B, and C -the points cut by the three lines of sight corresponding to three wires A’, B’ and C’ - top, bottom and axial hairs of the diaphragm i = interval b/w the stadia hairs (stadia interval) – length of imageAB = S = staff interceptF= focusf = focal length of the objectiveV = vertical axis of the instrumentv=distance between optical centre and imageu= distance between optical centre and staffd= distance between optical centre and vertical axis of instrument

S

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Similar triangle B’O’A’ and BO’A i/S = v/u v = iu/SProperties of Lens , 1/v + 1/u = 1/f Sub values of v, 1/(iu/S) + 1/u = 1/f S/iu + i/u = 1/f

1 S + 1 = 1u i f

u = S + 1 f i

D = u + d = S + 1 f + d i

D = (f/i) x S + (f+d) f/i – multiplying : f+d – additive constant


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Horizontal Sight

• A, C, and B = the points cut by the three


O is the optical centre of the objective of an external focusing telescopeA, B and C = the points cut by the three lines of sight corresponding to three wires a, b and c = bottom, top and hairs of the diaphragmab = i = interval b/w the stadia hairs (stadia interval)AB = s = staff intercept; f = focal length of the objective


f1 = horizontal distance of the staff from the optical centre of the objectivef2 = horizontal distance of the cross-wires from O.d = distance of the vertical axis of the instrument from O.D = horizontal distance of the staff from the vertical axis of the instruments. M = centre of the instrument, corresponding to the vertical axis.

Since the rays BOb and AOa pass through the optical centre, they are straight so that AOB and aOb are similar. Hence,

f1 sf2 i

=

Again, since f1 and f2 are conjugate focal distances, we have from lens formula,

1 1 1f f1 f2

+=

Multiplying throughout by ff1, we get f1 = f + ff1 f2Substituting the values of in the above, we get

f1 sf2 i

=

f1 = f + fsi

Horizontal distance between the axis and the staff is D = f1 + d

D = s + (f + d) = k . s + Cfi


Equation is known as the distance equation. In order to get the horizontal distance, therefore, the staff intercept s is to be found by subtracting the staff readings corresponding to the top and bottom stadia hairs.

Determination of constant K and C

1st method:

In this method, the additive constant C = (f + d) is measured from the instrument while the multiplying constant k is computed from field observations:

1. Focus the instrument to a distant object and measure along the telescope the distance between the objective and cross-hairs,

2. The distance d between the instrument axis and the objective is variable in the case of external focusing telescope, being greater for short sights and smaller for long sights. It should, therefore be measured for average sight. Thus, the additive constant (f + d) is known.

1 1 1f f1 f2

+=

UNIT I Tacheometric Surveying3. To calculate the multiplying constant k, measure a known distance D1 and take the intercept s1 on the staff kept at that point, the line of sight being horizontal. Using the equation,

D1 = ks1 + C or k = For average value, staff intercepts, s2, s3 etc., can be measured

corresponding to distance D2, D3 etc., and mean value can be calculated. Note: In case of some external focusing instruments, the eye-piece-diaphragm unit moves during focusing. For such instruments d is constant and does not vary while focusing.

D1 – C s

2nd method:

In this method, both the constants are determined by field observations as under:

1. Measure a line, about 200m long, on fairly level ground and drive pegs at some interval, say 50 meters.

2. Keep the staff on the pegs and observe the corresponding staff intercepts with horizontal sight.

3. Knowing the values of D and s for different points, a number of simultaneous equations can be formed by substituting the values of D and s in equation D = k.s + C. The simultaneous solution of successive pairs will give the values of k and C, and the average of these can be found.


For example, if s1 is the staff intercept corresponding to distance D1 and s2 corresponding to D2 we have,

D1 = k.s1 + C . . . . . (i) and D2 = k. s2 + C . . . . . (ii)

Subtracting (i) from (ii), we get

k = D2 – D1

s2 – s1

. . . . . . . . . (1)

Substituting the values of k in (i), we get

C = D1 - s1D2 – D1

s2 – s1

=D1s2 – D2s1

s2 – s1

. . . . . . . . . (2)

Thus equation (1) and (2) give the values of k and C.

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FIXED HAIR METHOD OF STADIA SYSTEM- Distance between the stadia hair is fixed- Distance between the station and staff = staff intercept x stadia constants

Methods to find Stadia Constants1. Line of sight is horizontal and staff vertical2. Line of sight inclined upwards and staff vertical3. Line of Sight inclined upwards and staff normal4. Line of Sight inclined downwards with staff vertical5. Line of sight inclined downwards and staff normal


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1. Line of Sight Horizontal and Staff Vertical

General Tacheometric equation : =100 & = 0RL of Staff station, P = Hi – hWhere as Hi = RL of BM + BS

BS = Back Sighth = central hair readingSivapriya Vijayasimhan

BS

D

O

O’ S

h

Height of Instrument (Hi)

BMP


C

A

O’

DO

A`

B

C`

P

P’

h

V

Ө

α

L

2.Line of sight inclined upwards and staff vertical

α

S

Line of axis

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Ois the optical centre of the objective of an external focusing telescopeA, B and C = the positions of staff corresponding to the cut points of the stadia and central hairsS= AC = staff intercepth=central hair readingV=vertical distance between instrument axis central hairD=horizontal distance between instrument and staffL=inclined distance between instrument axis and Bθ = angle of elevationα = angle made by outer and inner rays with central ray

A’C’ is drawn perpendicular to central ray, O’BL = +

Consider triangle ABA’ and CBC’ ABA’ = CBC’ = AA’B = 90o + α BC’C = 90o –α hence α is very small, it is taken as zero

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AA’B and BC’C = 900

So, AC’ = AC cos = S cos

Then

x +



C

A

O’

DO

A`

B

C`

P

P’

h

V

Ө

α

L

3.Line of sight inclined upwards and staff normal

α

S

Line of axis

h c

os

θ

h sin θ

L cos θ

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Vertical height of central hair = h cos θHorizontal distance between O and B = L cos θHorizontal distance, PP’ = h sinθ

Since staff is normal to line of collimation,

Horizontal distance, +h sin + +

Vertical Distance,


RL of staff station, P = (RL of instrument axis) +V – h cos

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UNIT I Tacheometric Surveying4.Case IV: Line of Sight Inclined Downwards with staff vertical


D Line of Axis

A B C’ C

V

h

P’

P

O

θO’

sin)(2

2Sin Sx

cos)(cosx 2

dfi

fV

dfSi

fD

RL of Staff P = (RL of axis of instrument) – V- h

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UNIT I Tacheometric Surveying5.Case V: Line of Sight Inclined Downwards with staff normal


Line of Axis

A B C

V

h cosθh

P P1

O

θO’

RL of Staff P = (RL of axis of instrument) – V- h cos θ

DL cos θ

L

h sinθ

sin)(sin

sin

sincos)(cos

)(

dfSi

fV

LV

hdfSi

fD

dfiSi

fL

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UNIT I Tacheometric SurveyingMovable Hair method of Stadia SystemPrinciple- Distance between stadia wires varies: staff intercept is constant- Staff has two tangents at known distance and third target at middleInstrument- Theodolite +subtense diaphragm = Subtense Theodolite- Upper and lower stadia wires can moved in vertical plane by using micrometer screws

- Distance = Turns of micrometer screws- Complete turns is read on scale and fractional parts on top and bottom eye piece- Sum of micrometer readings = total distance moved by stadia wiresSivapriya Vijayasimhan

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Observation- Middle target is bisected by central fixed hair- Micrometer screws are operated to move stadia wire up and down- Upper and lower targets are bisected by top and bottom wires

1.Line of Sight is horizontal

Where,C – constant varying from 600 to 1000n – number of readings in micrometerS – staff intercept (distance b/w upper and lower targets)

2.Line of Sight is inclined


d)(fx n

CSD

d)cos(f

cos2

n

CSD

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Subtense BarInstrument used to measure horizontal distance between instrument and a point on ground

Instrument• Theodolite – ordinary transit theodolite• Subtense bar made of metal of varying length 3 to 4 m• The bar can be locked in position by clamping screws • The bar can be levelled with the help of circular sprit level on the top. • At the mid point of the bar, a telescope arrangement or a sight rule with pair of vanes is

provided to align the bar perpendicular to the line of sight • Two targets are placed on the either ends of the bar such that they are equidistant from

the mid point • No staff is needed


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Alidade: line of sight perpendicular to the axis of the bar

Target

3 to 4 m

Spirit LevelTarget

Tripod

Alidade

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Procedure

BAC is measured by method of repetition , θAP is perpendicular to BC and bisects P

Note: –ve error in measurement of θ produce +ve error in D and vice-versa


A P

C

B

S

D

2cot

2

122

tan

2/

2tan,

SD

SD

D

SBAC

θ

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If an error of δθ ( -ve) will cause an error of δD (+ve)

If an error of δθ ( +ve) will cause an error of δD (-ve)


D

D

D

D

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Tangential System Of Tacheometry- No stadia hairs- Levelling staff with vanes or targets at known distance- Horizontal and vertical distances are measured by measuring the angles of

elevation or depression

Methods Case I : Both Angles of target are Angles of elevationCase II : Both angles of target are Angles of DepressionCase III : One angle is angle of elevation and the other is angle of depression


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Case I : Both Angles of target are Angles of elevation


O’

O

S

h

V

B

A

D

C1

C2

θ2θ1

O’ -Instrument axisO – Instrument stationC1 – Staff stationV – vertical distance between lower vane and axis of instrumentS – distance between the targetsθ1 - vertical angle by upper targetsθ2 - vertical angle lower targetsh – height of lower vane above the staff station

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21

2

21

21

2

1

tantan

tan

tantan

)tan(tan

tan

tan

SV

SD

DS

DV

DSV

21

2

21

21

2

1

tantan

tan

tantan

)tan(tan

tan

tan

SV

SD

DS

DV

DSV

RL of station C1 = RL of instrument axis + V - h

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Case II : Both Angles of target are Angles of depression


V

S

h

θ1 θ2

C2

A

B

C1

O’

O

D

12

2

12

12

1

2

tantan

tan

tantan

)tan(tan

tan

tan

SV

SD

DS

DSV

DV

RL of station A = RL of instrument axis - V - h

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Case III : One angle is angle of elevation and the other is angle of depression


SV

h

θ1θ2

C2

C1

O’

O

D

21

2

21

1

2

tantan

tan

tantan

tan

tan

SV

SD

DVS

DV

RL of station A = RL of instrument axis - V - h

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Anallatic Lens- Convex lens between the object glass and diaphragm to make additive constant as zero- Reduces the brilliance of image- Distance = difference in stadia hair x multiplying constant (100)

DK

d

S

mf2 f1

V P

b2

a2

A

b

i’

a

b3

a3

f'a1

b1

N

N’

A

B

Oi

O – optical centre of object glass: A – optical centre of anallatic lensP – principal focus of anallatic lensK – distance between object glass and anallatic lensD – distance between vertical axis of instrument and staffS - staff interceptv – vertical axis of the instrument

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UNIT I Tacheometric Surveyingf - focal length of the objectivef1 and f2 – conjugate focal lengths of object glassf' – focal lengths of anallatic lensd - distance between optical centre and vertical axis of instrumentm - distance between optical centre and real image , abi – length of image a3b3 when anallatic lens is not providedi ‘– length of image a3b3 when anallatic lens is provided

Ray of light from AB along AN and BN’, meets at PP –principal focus of anallatic lensDiverging ray from P emerge direction parallel to axis of telescope after passing through

anallatic lensReal image “ab” is formed

Without inclusion of Anallatic lens, law of lenses


2

1

21

111

f

f

i

S

fff

1

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With inclusion of Anallatic lens, law of lenses

With inclusion of anallatic lens, imaginary object a2b2 is seen.

Final image “ab” is formed in stadia hairs

Eliminate m,f2 and I, from equation 1 and 2;


Km

Kf

i

i

KmKff

2

1'

'

111

1''

)('

1

1

1

2

2

1

12

11

12

i

i

f

Kf

Kfi

i

KmSubstitute

ff

fff

ff

Sf

ff

Si

if

S

f

in equation 2

2

3

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Substitute values of i and f2 from equation 3 , we get

df

fdfK

Kff

fKfd

dKff

fKf

Kffi

SffD

dfDNow

Kff

fKf

Kffi

Sfff

'

0)'(

)'(

)'(

)'(

)'('

'

,

)'(

)'(

)'('

'

1

1

D is proportional to S

By adopting suitable values for f,f’,K and i; K’ is derived as follows

100)'('

''

Kffi

ffK

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Errors and Precautions in Tacheometric Surveying1. Errors of observation2. Instrument Errors3. Errors due to natural causes

1.Errors of observation


Error PrecautionsIncorrect centering and levelling Proper centering and levelling of plate bubble and altitude

should be taken careVerticality of staff Plumb bob is used to check the verticalityParallax error Proper focusing before starting of workDistance of station beyond the scope of telescope

Graduations on staff are clearly and distinctly seen

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2. Instrument Errors

3. Errors due to natural causes


Error PrecautionsHigh wind: staff and instrument subject to variation

Work should be suspended or temporary barear used to some extended

Hot weather: Tacheometer subjected to expansion

Readings taken under some shade

Hot weather: Poor visibility of staff

No direct sunlight on object glass

Error PrecautionsAdjustments in Tacheometer Checked and rectified before starting Graduation of staff or stadia hair Checked and corrected or replacedMultiplying constant ≠0 Field test should be conducted inorder to avoid constant errors

tacheometric surveying. unit i tacheometric surveying contents tacheometric surveying tangential,...

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